Math核心计算通关讲义

核心计算通关讲义

核心计算通关讲义

第一组 含有x的多项式的积分（不带根号）

  1. ∫1ax+bdx=1aln⁡∣ax+b∣+C\int \frac{1}{a x+b} d x=\frac{1}{a} \ln |a x+b|+C
  2. ∫(ax+b)μdx=1a(μ+1)(ax+b)μ+1+C,μ≠−1\int(a x+b)^{\mu} d x=\frac{1}{a(\mu+1)}(a x+b)^{\mu+1}+C, \mu \neq-1
  3. ∫xax+bdx=1a2(ax−bln⁡∣ax+b∣)+C\int \frac{x}{a x+b} d x=\frac{1}{a^{2}}(a x-b \ln |a x+b|)+C
  4. ∫dxx(ax+b)=−1bln⁡∣ax+bx∣+C\int \frac{d x}{x(a x+b)}=-\frac{1}{b} \ln \left|\frac{a x+b}{x}\right|+C
  5. ∫x(ax+b)2dx=1a2(ln⁡∣ax+b∣+bax+b)+C\int \frac{x}{(a x+b)^{2}} d x=\frac{1}{a^{2}}\left(\ln |a x+b|+\frac{b}{a x+b}\right)+C
  6. ∫dxax2+b={1abarctan⁡abx+C,a>0,b>0,12−abln⁡∣ax−−bax+−b∣+C,a>0,blt;0\int \frac{d x}{a x^{2}+b}= \begin{cases}\frac{1}{\sqrt{a b}} \arctan \sqrt{\frac{a}{b}} x+C, & a>0, b>0, \\ \frac{1}{2 \sqrt{-a b}} \ln \left|\frac{\sqrt{a} x-\sqrt{-b}}{\sqrt{a} x+\sqrt{-b}}\right|+C, & a>0, b<0\end{cases}
  7. ∫xax2+bdx=12aln⁡∣ax2+b∣+C\int \frac{x}{a x^{2}+b} d x=\frac{1}{2 a} \ln \left|a x^{2}+b\right|+C
  8. ∫dx1−x2=12ln⁡∣1+x1−x∣+C\int \frac{d x}{1-x^{2}}=\frac{1}{2} \ln \left|\frac{1+x}{1-x}\right|+C
  9. ∫x1+x2dx=12ln⁡(1+x2)+C\int \frac{x}{1+x^{2}} d x=\frac{1}{2} \ln \left(1+x^{2}\right)+C
  10. ∫dx1+x3=13ln⁡∣1+x∣1−x+x2+13arctan⁡2x−13+C\int \frac{d x}{1+x^{3}}=\frac{1}{3} \ln \frac{|1+x|}{\sqrt{1-x+x^{2}}}+\frac{1}{\sqrt{3}} \arctan \frac{2 x-1}{\sqrt{3}}+C
  11. ∫x1+x3dx=16ln⁡x2−x+1(1+x)2+13arctan⁡2x−13+C\int \frac{x}{1+x^{3}} d x=\frac{1}{6} \ln \frac{x^{2}-x+1}{(1+x)^{2}}+\frac{1}{\sqrt{3}} \arctan \frac{2 x-1}{\sqrt{3}}+C
  12. ∫dx1+x4=24arctan⁡2(x2−1)2x−28ln⁡x2−2x+1x2+2x+1+C\int \frac{d x}{1+x^{4}}=\frac{\sqrt{2}}{4} \arctan \frac{\sqrt{2}\left(x^{2}-1\right)}{2 x}-\frac{\sqrt{2}}{8} \ln \frac{x^{2}-\sqrt{2} x+1}{x^{2}+\sqrt{2} x+1}+C
  13. ∫dx1−x4=14ln⁡∣1+x1−x∣+12arctan⁡x+C\int \frac{d x}{1-x^{4}}=\frac{1}{4} \ln \left|\frac{1+x}{1-x}\right|+\frac{1}{2} \arctan x+C
  14. ∫x1−x3dx=16ln⁡1+x+x2(1−x)2−13arctan⁡2x+13+C\int \frac{x}{1-x^{3}} d x=\frac{1}{6} \ln \frac{1+x+x^{2}}{(1-x)^{2}}-\frac{1}{\sqrt{3}} \arctan \frac{2 x+1}{\sqrt{3}}+C
  15. ∫x1+x4dx=12arctan⁡x2+C\int \frac{x}{1+x^{4}} d x=\frac{1}{2} \arctan x^{2}+C
  16. ∫x1−x4dx=14ln⁡∣1+x21−x2∣+C\int \frac{x}{1-x^{4}} d x=\frac{1}{4} \ln \left|\frac{1+x^{2}}{1-x^{2}}\right|+C
  17. ∫dxc2+x2=1carctan⁡xc+C\int \frac{d x}{c^{2}+x^{2}}=\frac{1}{c} \arctan \frac{x}{c}+C
  18. ∫dx(c2+x2)2=12c3(cxc2+x2+arctan⁡xc)+C\int \frac{d x}{\left(c^{2}+x^{2}\right)^{2}}=\frac{1}{2 c^{3}}\left(\frac{c x}{c^{2}+x^{2}}+\arctan \frac{x}{c}\right)+C
  19. ∫x(c2+x2)2dx=−12(c2+x2)+C\int \frac{x}{\left(c^{2}+x^{2}\right)^{2}} d x=-\frac{1}{2\left(c^{2}+x^{2}\right)}+C
  20. ∫x2c2+x2dx=x−carctan⁡xc+C\int \frac{x^{2}}{c^{2}+x^{2}} d x=x-c \arctan \frac{x}{c}+C
  21. ∫x2(c2+x2)2dx=−x2(c2+x2)+12carctan⁡xc+C\int \frac{x^{2}}{\left(c^{2}+x^{2}\right)^{2}} d x=-\frac{x}{2\left(c^{2}+x^{2}\right)}+\frac{1}{2 c} \arctan \frac{x}{c}+C
  22. ∫x3c2+x2dx=x22−c22ln⁡(c2+x2)+C\int \frac{x^{3}}{c^{2}+x^{2}} d x=\frac{x^{2}}{2}-\frac{c^{2}}{2} \ln \left(c^{2}+x^{2}\right)+C
  23. ∫x3(c2+x2)2dx=c22(c2+x2)+12ln⁡(c2+x2)+C\int \frac{x^{3}}{\left(c^{2}+x^{2}\right)^{2}} d x=\frac{c^{2}}{2\left(c^{2}+x^{2}\right)}+\frac{1}{2} \ln \left(c^{2}+x^{2}\right)+C
  24. ∫dxx(c2+x2)=12c2ln⁡x2c2+x2+C\int \frac{d x}{x\left(c^{2}+x^{2}\right)}=\frac{1}{2 c^{2}} \ln \frac{x^{2}}{c^{2}+x^{2}}+C
  25. ∫dxx(c2+x2)2=12c2(c2+x2)+12c4ln⁡x2c2+x2+C\int \frac{d x}{x\left(c^{2}+x^{2}\right)^{2}}=\frac{1}{2 c^{2}\left(c^{2}+x^{2}\right)}+\frac{1}{2 c^{4}} \ln \frac{x^{2}}{c^{2}+x^{2}}+C
  26. ∫dxx2(c2+x2)=−1c2x−1c3arctan⁡xc+C\int \frac{d x}{x^{2}\left(c^{2}+x^{2}\right)}=-\frac{1}{c^{2} x}-\frac{1}{c^{3}} \arctan \frac{x}{c}+C
  27. ∫xx2−c2dx=12ln⁡∣x2−c2∣+C\int \frac{x}{x^{2}-c^{2}} d x=\frac{1}{2} \ln \left|x^{2}-c^{2}\right|+C
  28. ∫dxx2−c2=12cln⁡∣x−cx+c∣+C\int \frac{d x}{x^{2}-c^{2}}=\frac{1}{2 c} \ln \left|\frac{x-c}{x+c}\right|+C
  29. ∫dx(c2−x2)2=x2c2(c2−x2)+14c3ln⁡∣c+xc−x∣+C\int \frac{d x}{\left(c^{2}-x^{2}\right)^{2}}=\frac{x}{2 c^{2}\left(c^{2}-x^{2}\right)}+\frac{1}{4 c^{3}} \ln \left|\frac{c+x}{c-x}\right|+C
  30. ∫x2c2−x2dx=−x+c2ln⁡∣c+xc−x∣+C\int \frac{x^{2}}{c^{2}-x^{2}} d x=-x+\frac{c}{2} \ln \left|\frac{c+x}{c-x}\right|+C
  31. ∫x2(c2−x2)2dx=x2(c2−x2)−14cln⁡∣c+xc−x∣+C\int \frac{x^{2}}{\left(c^{2}-x^{2}\right)^{2}} d x=\frac{x}{2\left(c^{2}-x^{2}\right)}-\frac{1}{4 c} \ln \left|\frac{c+x}{c-x}\right|+C
  32. ∫dxx2(c2−x2)=−1c2x+12c3ln⁡∣c+xc−x∣+C\int \frac{d x}{x^{2}\left(c^{2}-x^{2}\right)}=-\frac{1}{c^{2} x}+\frac{1}{2 c^{3}} \ln \left|\frac{c+x}{c-x}\right|+C
  33. ∫ax+bdx=23a(ax+b)3+C\int \sqrt{a x+b} d x=\frac{2}{3 a} \sqrt{(a x+b)^{3}}+C
  34. ∫ax+bx2dx=−ax+bx+a2∫dxxax+b\int \frac{\sqrt{a x+b}}{x^{2}} d x=-\frac{\sqrt{a x+b}}{x}+\frac{a}{2} \int \frac{d x}{x \sqrt{a x+b}}
  35. ∫dxxax+b={1bln⁡∣ax+b−bax+b+b∣+C,b>0,2−barctan⁡ax+b−b+C,blt;0\int \frac{d x}{x \sqrt{a x+b}}= \begin{cases}\frac{1}{\sqrt{b}} \ln \left|\frac{\sqrt{a x+b}-\sqrt{b}}{\sqrt{a x+b}+\sqrt{b}}\right|+C, & b>0, \\ \frac{2}{\sqrt{-b}} \arctan \sqrt{\frac{a x+b}{-b}}+C, & b<0\end{cases}
  36. ∫xx2−a2dx=x2−a2+C\int \frac{x}{\sqrt{x^{2}-a^{2}}} d x=\sqrt{x^{2}-a^{2}}+C
  37. ∫x(x2−a2)3dx=−1x2−a2+C\int \frac{x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}} d x=-\frac{1}{\sqrt{x^{2}-a^{2}}}+C
  38. ∫x2x2−a2dx=x2x2−a2+a22ln⁡∣x+x2−a2∣+C\int \frac{x^{2}}{\sqrt{x^{2}-a^{2}}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C
  39. ∫dxxx2−a2=1aarccos⁡a∣x∣+C\int \frac{d x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \arccos \frac{a}{|x|}+C
  40. ∫dxx2x2−a2=x2−a2a2x+C\int \frac{d x}{x^{2} \sqrt{x^{2}-a^{2}}}=\frac{\sqrt{x^{2}-a^{2}}}{a^{2} x}+C
  41. ∫xx2−a2dx=13(x2−a2)3+C\int x \sqrt{x^{2}-a^{2}} d x=\frac{1}{3} \sqrt{\left(x^{2}-a^{2}\right)^{3}}+C
  42. ∫x2−a2xdx=x2−a2−aarccos⁡a∣x∣+C\int \frac{\sqrt{x^{2}-a^{2}}}{x} d x=\sqrt{x^{2}-a^{2}}-a \arccos \frac{a}{|x|}+C

第二组 含有x的多项式的积分（带根号）

  1. ∫ax+bdx=23a(ax+b)3+C\int \sqrt{a x+b} d x=\frac{2}{3 a} \sqrt{(a x+b)^{3}}+C
  2. ∫ax+bx2dx=−ax+bx+a2∫dxxax+b\int \frac{\sqrt{a x+b}}{x^{2}} d x=-\frac{\sqrt{a x+b}}{x}+\frac{a}{2} \int \frac{d x}{x \sqrt{a x+b}}
  3. ∫dxxax+b={1bln⁡∣ax+b−bax+b+b∣+C,b>0,2−barctan⁡ax+b−b+C,blt;0\int \frac{d x}{x \sqrt{a x+b}}= \begin{cases}\frac{1}{\sqrt{b}} \ln \left|\frac{\sqrt{a x+b}-\sqrt{b}}{\sqrt{a x+b}+\sqrt{b}}\right|+C, & b>0, \\ \frac{2}{\sqrt{-b}} \arctan \sqrt{\frac{a x+b}{-b}}+C, & b<0\end{cases}
  4. ∫xx2−a2dx=x2−a2+C\int \frac{x}{\sqrt{x^{2}-a^{2}}} d x=\sqrt{x^{2}-a^{2}}+C
  5. ∫x(x2−a2)3dx=−1x2−a2+C\int \frac{x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}} d x=-\frac{1}{\sqrt{x^{2}-a^{2}}}+C
  6. ∫x2x2−a2dx=x2x2−a2+a22ln⁡∣x+x2−a2∣+C\int \frac{x^{2}}{\sqrt{x^{2}-a^{2}}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C
  7. ∫dxxx2−a2=1aarccos⁡a∣x∣+C\int \frac{d x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \arccos \frac{a}{|x|}+C
  8. ∫dxx2x2−a2=x2−a2a2x+C\int \frac{d x}{x^{2} \sqrt{x^{2}-a^{2}}}=\frac{\sqrt{x^{2}-a^{2}}}{a^{2} x}+C
  9. ∫xx2−a2dx=13(x2−a2)3+C\int x \sqrt{x^{2}-a^{2}} d x=\frac{1}{3} \sqrt{\left(x^{2}-a^{2}\right)^{3}}+C
  10. ∫x2−a2xdx=x2−a2−aarccos⁡a∣x∣+C\int \frac{\sqrt{x^{2}-a^{2}}}{x} d x=\sqrt{x^{2}-a^{2}}-a \arccos \frac{a}{|x|}+C
  11. ∫x2−a2x2dx=−x2−a2x+ln⁡∣x+x2−a2∣+C\int \frac{\sqrt{x^{2}-a^{2}}}{x^{2}} d x=-\frac{\sqrt{x^{2}-a^{2}}}{x}+\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C
  12. ∫dx(a2−x2)3=xa2a2−x2+C\int \frac{d x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}=\frac{x}{a^{2} \sqrt{a^{2}-x^{2}}}+C
  13. ∫xa2−x2dx=−a2−x2+C\int \frac{x}{\sqrt{a^{2}-x^{2}}} d x=-\sqrt{a^{2}-x^{2}}+C
  14. ∫x(a2−x2)3dx=1a2−x2+C\int \frac{x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}} d x=\frac{1}{\sqrt{a^{2}-x^{2}}}+C
  15. ∫x2a2−x2dx=−x2a2−x2+a22arcsin⁡xa+C\int \frac{x^{2}}{\sqrt{a^{2}-x^{2}}} d x=-\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \arcsin \frac{x}{a}+C
  16. ∫x2(a2−x2)3dx=xa2−x2−arcsin⁡xa+C\int \frac{x^{2}}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}} d x=\frac{x}{\sqrt{a^{2}-x^{2}}}-\arcsin \frac{x}{a}+C
  17. ∫dxx2a2−x2=−a2−x2a2x+C\int \frac{d x}{x^{2} \sqrt{a^{2}-x^{2}}}=-\frac{\sqrt{a^{2}-x^{2}}}{a^{2} x}+C
  18. ∫xa2−x2dx=−13(a2−x2)3+C\int x \sqrt{a^{2}-x^{2}} d x=-\frac{1}{3} \sqrt{\left(a^{2}-x^{2}\right)^{3}}+C
  19. ∫a2−x2x2dx=−a2−x2x−arcsin⁡xa+C\int \frac{\sqrt{a^{2}-x^{2}}}{x^{2}} d x=-\frac{\sqrt{a^{2}-x^{2}}}{x}-\arcsin \frac{x}{a}+C
  20. ∫(a2−x2)3dx=14[x(a2−x2)3+3a2x2a2−x2+3a42arcsin⁡xa]+C\int \sqrt{\left(a^{2}-x^{2}\right)^{3}} d x=\frac{1}{4}\left[x \sqrt{\left(a^{2}-x^{2}\right)^{3}}+\frac{3 a^{2} x}{2} \sqrt{a^{2}-x^{2}}+\frac{3 a^{4}}{2} \arcsin \frac{x}{a}\right]+C
  21. ∫x2a2−x2dx=−x4(a2−x2)3+a28(xa2−x2+a2arcsin⁡xa)+C\int x^{2} \sqrt{a^{2}-x^{2}} d x=-\frac{x}{4} \sqrt{\left(a^{2}-x^{2}\right)^{3}}+\frac{a^{2}}{8}\left(x \sqrt{a^{2}-x^{2}}+a^{2} \arcsin \frac{x}{a}\right)+C
  22. ∫2ax−x2dx=12[(x−a)2ax−x2+a2arcsin⁡x−aa]+C\int \sqrt{2 a x-x^{2}} d x=\frac{1}{2}\left[(x-a) \sqrt{2 a x-x^{2}}+a^{2} \arcsin \frac{x-a}{a}\right]+C
  23. ∫dx2ax−x2=−arccos⁡x−aa+C=arcsin⁡x−aa+C\int \frac{d x}{\sqrt{2 a x-x^{2}}}=-\arccos \frac{x-a}{a}+C=\arcsin \frac{x-a}{a}+C
  24. ∫dx2ax+x2=ln⁡∣x+a+2ax+x2∣+C\int \frac{d x}{\sqrt{2 a x+x^{2}}}=\ln \left|x+a+\sqrt{2 a x+x^{2}}\right|+C
  25. ∫dxxxn+a2=2naln⁡xn+a2−axn+C\int \frac{d x}{x \sqrt{x^{n}+a^{2}}}=\frac{2}{n a} \ln \frac{\sqrt{x^{n}+a^{2}}-a}{\sqrt{x^{n}}}+C
  26. ∫dxxxn−a2=−2naarcsin⁡axn+C\int \frac{d x}{x \sqrt{x^{n}-a^{2}}}=-\frac{2}{n a} \arcsin \frac{a}{\sqrt{x^{n}}}+C
  27. ∫xa3−x3dx=23arcsin⁡(xa)32+C\int \sqrt{\frac{x}{a^{3}-x^{3}}} d x=\frac{2}{3} \arcsin \left(\frac{x}{a}\right)^{\frac{3}{2}}+C
  28. ∫1+x1−xdx=arcsin⁡x−1−x2+C\int \sqrt{\frac{1+x}{1-x}} d x=\arcsin x-\sqrt{1-x^{2}}+C
  29. ∫a−bxa+bxdx=1b(a+bx)(a−bx)+abarcsin⁡bxa+C\int \sqrt{\frac{a-b x}{a+b x}} d x=\frac{1}{b} \sqrt{(a+b x)(a-b x)}+\frac{a}{b} \arcsin \frac{b x}{a}+C
  30. ∫dxx2+a2=ln⁡(x+x2+a2)+C\int \frac{d x}{\sqrt{x^{2}+a^{2}}}=\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C
  31. ∫dx(x2+a2)3=xa2x2+a2+C\int \frac{d x}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}}=\frac{x}{a^{2} \sqrt{x^{2}+a^{2}}}+C
  32. ∫xx2+a2dx=x2+a2+C\int \frac{x}{\sqrt{x^{2}+a^{2}}} d x=\sqrt{x^{2}+a^{2}}+C
  33. ∫x(x2+a2)3dx=−1x2+a2+C\int \frac{x}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}} d x=-\frac{1}{\sqrt{x^{2}+a^{2}}}+C
  34. ∫x2x2+a2dx=x2x2+a2−a22ln⁡(x+x2+a2)+C\int \frac{x^{2}}{\sqrt{x^{2}+a^{2}}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}-\frac{a^{2}}{2} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C
  35. ∫dxx2x2+a2=−x2+a2a2x+C\int \frac{d x}{x^{2} \sqrt{x^{2}+a^{2}}}=-\frac{\sqrt{x^{2}+a^{2}}}{a^{2} x}+C
  36. ∫x2+a2dx=x2x2+a2+a22ln⁡(x+x2+a2)+C\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C
  37. ∫x2+a2x2dx=−x2+a2x+ln⁡(x+x2+a2)+C\int \frac{\sqrt{x^{2}+a^{2}}}{x^{2}} d x=-\frac{\sqrt{x^{2}+a^{2}}}{x}+\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C
  38. ∫xx2−c2dx=12ln⁡∣x2−c2∣+C\int \frac{x}{x^{2}-c^{2}} d x=\frac{1}{2} \ln \left|x^{2}-c^{2}\right|+C
  39. ∫a+xa3+x3dx=23aarctan⁡2x−a3a+C\int \frac{a+x}{a^{3}+x^{3}} d x=\frac{2}{\sqrt{3} a} \arctan \frac{2 x-a}{\sqrt{3} a}+C
  40. ∫a−xa3−x3dx=23aarctan⁡2x+a3a+C\int \frac{a-x}{a^{3}-x^{3}} d x=\frac{2}{\sqrt{3} a} \arctan \frac{2 x+a}{\sqrt{3} a}+C
  41. ∫dxax2+bx+c={1aln⁡∣2ax+b+2aax2+bx+c∣+C,b2−4ac≠0,1aln⁡∣x+b2a∣+C,b2−4ac=0\int \frac{d x}{\sqrt{a x^{2}+b x+c}}= \begin{cases}\frac{1}{\sqrt{a}} \ln \left|2 a x+b+2 \sqrt{a} \sqrt{a x^{2}+b x+c}\right|+C, & b^{2}-4 a c \neq 0, \\ \frac{1}{\sqrt{a}} \ln \left|x+\frac{b}{2 a}\right|+C, & b^{2}-4 a c=0\end{cases}
  42. ∫dxc+bx−ax2=1aarcsin⁡2ax−bb2+4ac+C\int \frac{d x}{\sqrt{c+b x-a x^{2}}}=\frac{1}{\sqrt{a}} \arcsin \frac{2 a x-b}{\sqrt{b^{2}+4 a c}}+C
  43. ∫x−ax−bdx=(x−b)x−ax−b+(b−a)ln⁡(∣x−a∣+∣x−b∣)+C\int \sqrt{\frac{x-a}{x-b}} d x=(x-b) \sqrt{\frac{x-a}{x-b}}+(b-a) \ln (\sqrt{|x-a|}+\sqrt{|x-b|})+C
  44. ∫x−ab−xdx=(x−b)x−ab−x+(b−a)arctan⁡x−ab−x+C\int \sqrt{\frac{x-a}{b-x}} d x=(x-b) \sqrt{\frac{x-a}{b-x}}+(b-a) \arctan \sqrt{\frac{x-a}{b-x}}+C
  45. ∫dx(x−a)(b−x)=2arcsin⁡x−ab−a+C,a<b\int \frac{d x}{\sqrt{(x-a)(b-x)}}=2 \arcsin \sqrt{\frac{x-a}{b-a}}+C, a<b
  46. ∫(x−a)(b−x)dx=2x−a−b4⋅(x−a)(b−x)+(b−a)24arcsin⁡x−ab−a+C,a<b\int \sqrt{(x-a)(b-x)} d x=\frac{2 x-a-b}{4} \cdot \sqrt{(x-a)(b-x)}+\frac{(b-a)^{2}}{4} \arcsin \sqrt{\frac{x-a}{b-a}}+C, a<b

第三组 含有三角函数的有理式的积分（不带根号）

∫cos⁡3axdx=sin⁡axa−sin⁡3ax3a+C.\int \cos^3 ax dx = \frac{\sin ax}{a} - \frac{\sin^3 ax}{3a} + C.

∫tan⁡2axdx=1atan⁡ax−x+C.\int \tan^2 ax dx = \frac{1}{a}\tan ax - x + C.

∫tan⁡3axdx=12atan⁡2ax+1aln⁡∣cos⁡ax∣+C.\int \tan^3 ax dx = \frac{1}{2a}\tan^2 ax + \frac{1}{a}\ln|\cos ax| + C.

∫cot⁡2axdx=−1acot⁡ax−x+C.\int \cot^2 ax dx = -\frac{1}{a}\cot ax - x + C.

∫cot⁡3axdx=−12acot⁡2ax−1aln⁡∣sin⁡ax∣+C.\int \cot^3 ax dx = -\frac{1}{2a}\cot^2 ax - \frac{1}{a}\ln|\sin ax| + C.

∫dxsin⁡3ax=−cos⁡ax2asin⁡2ax+12aln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\sin^3 ax} = -\frac{\cos ax}{2a\sin^2 ax} + \frac{1}{2a}\ln\left|\tan \frac{ax}{2}\right| + C.

∫dxcos⁡3ax=−sin⁡ax2acos⁡2ax+12aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{dx}{\cos^3 ax} = -\frac{\sin ax}{2a\cos^2 ax} + \frac{1}{2a}\ln\left|\tan\left(\frac{\pi}{4} + \frac{ax}{2}\right)\right| + C.

∫sin⁡mxsin⁡nxdx=sin⁡(m−n)x2(m−n)−sin⁡(m+n)x2(m+n)+C,m2≠n2.\int \sin mx \sin nx dx = \frac{\sin(m-n)x}{2(m-n)} - \frac{\sin(m+n)x}{2(m+n)} + C, \quad m^2 \neq n^2.

∫sin⁡mxcos⁡nxdx=−cos⁡(m−n)x2(m−n)−cos⁡(m+n)x2(m+n)+C,m2≠n2.\int \sin mx \cos nx dx = -\frac{\cos(m-n)x}{2(m-n)} - \frac{\cos(m+n)x}{2(m+n)} + C, \quad m^2 \neq n^2.

∫cos⁡mxcos⁡nxdx=sin⁡(m−n)x2(m−n)+sin⁡(m+n)x2(m+n)+C,m2≠n2.\int \cos mx \cos nx dx = \frac{\sin(m-n)x}{2(m-n)} + \frac{\sin(m+n)x}{2(m+n)} + C, \quad m^2 \neq n^2.

∫sin⁡axcos⁡axdx=12asin⁡2ax+C.\int \sin ax \cos ax dx = \frac{1}{2a} \sin^2 ax + C.

∫sin⁡2axcos⁡axdx=sin⁡3ax3a+C.\int \sin^2 ax \cos ax dx = \frac{\sin^3 ax}{3a} + C.

∫sin⁡2axcos⁡2axdx=−sin⁡4ax32a+x8+C.\int \sin^2 ax \cos^2 ax dx = -\frac{\sin 4ax}{32a} + \frac{x}{8} + C.

∫sin⁡2axcos⁡3axdx=13asin⁡3ax−15asin⁡5ax+C.\int \sin^2 ax \cos^3 ax dx = \frac{1}{3a} \sin^3 ax - \frac{1}{5a} \sin^5 ax + C.

∫sin⁡3axcos⁡axdx=sin⁡4ax4a+C.\int \sin^3 ax \cos ax dx = \frac{\sin^4 ax}{4a} + C.

∫sin⁡3axcos⁡2axdx=−cos⁡3ax3a+cos⁡5ax5a+C.\int \sin^3 ax \cos^2 ax dx = -\frac{\cos^3 ax}{3a} + \frac{\cos^5 ax}{5a} + C.

∫sin⁡3axcos⁡3axdx=−116acos⁡2ax+148acos⁡32ax+C.\int \sin^3 ax \cos^3 ax dx = -\frac{1}{16a} \cos 2ax + \frac{1}{48a} \cos^3 2ax + C.

∫sin⁡axcos⁡2axdx=1acos⁡ax+C=sec⁡ax+C.\int \frac{\sin ax}{\cos^2 ax} dx = \frac{1}{a \cos ax} + C = \sec ax + C.

∫sin⁡2axcos⁡3axdx=sin⁡ax2acos⁡2ax−12aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{\sin^2 ax}{\cos^3 ax} dx = \frac{\sin ax}{2a \cos^2 ax} - \frac{1}{2a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) \right| + C.

∫sin⁡2axcos⁡4axdx=tan⁡3ax3a+C.\int \frac{\sin^2 ax}{\cos^4 ax} dx = \frac{\tan^3 ax}{3a} + C.

∫sin⁡3axcos⁡axdx=−sin⁡2ax2a−1aln⁡∣cos⁡ax∣+C.\int \frac{\sin^3 ax}{\cos ax} dx = -\frac{\sin^2 ax}{2a} - \frac{1}{a} \ln |\cos ax| + C.

∫sin⁡3axcos⁡2axdx=cos⁡axa+1acos⁡ax+C.\int \frac{\sin^3 ax}{\cos^2 ax} dx = \frac{\cos ax}{a} + \frac{1}{a \cos ax} + C.

∫sin⁡3axcos⁡6axdx=15acos⁡5ax−13acos⁡3ax+C.\int \frac{\sin^3 ax}{\cos^6 ax} dx = \frac{1}{5a \cos^5 ax} - \frac{1}{3a \cos^3 ax} + C.

∫cos⁡2axsin⁡axdx=cos⁡axa+1aln⁡∣tan⁡(ax2)∣+C.\int \frac{\cos^2 ax}{\sin ax} dx = \frac{\cos ax}{a} + \frac{1}{a} \ln \left| \tan \left( \frac{ax}{2} \right) \right| + C.

∫cos⁡2axsin⁡3axdx=−cos⁡ax2asin⁡2ax−12aln⁡∣tan⁡(ax2)∣+C.\int \frac{\cos^2 ax}{\sin^3 ax} dx = -\frac{\cos ax}{2a \sin^2 ax} - \frac{1}{2a} \ln \left| \tan \left( \frac{ax}{2} \right) \right| + C.

∫cos⁡2axsin⁡4axdx=−cot⁡3ax3a+C.\int \frac{\cos^2 ax}{\sin^4 ax} dx = -\frac{\cot^3 ax}{3a} + C.

∫cos⁡3axsin⁡axdx=cos⁡2ax2a+1aln⁡∣sin⁡ax∣+C.\int \frac{\cos^3 ax}{\sin ax} dx = \frac{\cos^2 ax}{2a} + \frac{1}{a} \ln |\sin ax| + C.

∫cos⁡3axsin⁡2axdx=−sin⁡axa−1asin⁡ax+C.\int \frac{\cos^3 ax}{\sin^2 ax} dx = -\frac{\sin ax}{a} - \frac{1}{a \sin ax} + C.

∫cos⁡3axsin⁡4axdx=−13asin⁡3ax+1asin⁡ax+C.\int \frac{\cos^3 ax}{\sin^4 ax} dx = -\frac{1}{3a \sin^3 ax} + \frac{1}{a \sin ax} + C.

∫dxsin⁡axcos⁡ax=1aln⁡∣tan⁡ax∣+C.\int \frac{dx}{\sin ax \cos ax} = \frac{1}{a} \ln |\tan ax| + C.

∫dxsin⁡axcos⁡2ax=1a(sec⁡ax+ln⁡∣tan⁡ax2∣)+C.\int \frac{dx}{\sin ax \cos^2 ax} = \frac{1}{a} \left( \sec ax + \ln \left| \tan \frac{ax}{2} \right| \right) + C.

∫dxsin⁡2axcos⁡ax=−1acsc⁡ax+1aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{dx}{\sin^2 ax \cos ax} = -\frac{1}{a} \csc ax + \frac{1}{a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) \right| + C.

∫dxsin⁡2axcos⁡2ax=−2acot⁡2ax+C.\int \frac{dx}{\sin^2 ax \cos^2 ax} = -\frac{2}{a} \cot 2ax + C.

∫dxsin⁡2axcos⁡3ax=12asin⁡ax(1cos⁡2ax−3)+32aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{dx}{\sin^2 ax \cos^3 ax} = \frac{1}{2a \sin ax} \left( \frac{1}{\cos^2 ax} - 3 \right) + \frac{3}{2a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) \right| + C.

∫dxsin⁡2axcos⁡4ax=tan⁡3ax3a+tan⁡axa−2acot⁡2ax+C.\int \frac{dx}{\sin^2 ax \cos^4 ax} = \frac{\tan^3 ax}{3a} + \frac{\tan ax}{a} - \frac{2}{a} \cot 2ax + C.

∫dxsin⁡3axcos⁡ax=−12asin⁡2ax+1aln⁡∣tan⁡ax∣+C.\int \frac{dx}{\sin^3 ax \cos ax} = -\frac{1}{2a \sin^2 ax} + \frac{1}{a} \ln |\tan ax| + C.

∫dxsin⁡3axcos⁡2ax=1acos⁡ax−cos⁡ax2asin⁡2ax+32aln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\sin^3 ax \cos^2 ax} = \frac{1}{a \cos ax} - \frac{\cos ax}{2a \sin^2 ax} + \frac{3}{2a} \ln \left| \tan \frac{ax}{2} \right| + C.

∫dxsin⁡3axcos⁡3ax=−2cos⁡2axasin⁡22ax+2aln⁡∣tan⁡ax∣+C.\int \frac{dx}{\sin^3 ax \cos^3 ax} = -\frac{2 \cos 2ax}{a \sin^2 2ax} + \frac{2}{a} \ln |\tan ax| + C.

∫dxsin⁡3axcos⁡4ax=2acos⁡ax+13acos⁡3ax−cos⁡ax2asin⁡2ax+52aln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\sin^3 ax \cos^4 ax} = \frac{2}{a \cos ax} + \frac{1}{3a \cos^3 ax} - \frac{\cos ax}{2a \sin^2 ax} + \frac{5}{2a} \ln \left| \tan \frac{ax}{2} \right| + C.

∫dx1±sin⁡ax=∓1atan⁡(π4∓ax2)+C.\int \frac{dx}{1 \pm \sin ax} = \mp \frac{1}{a} \tan \left( \frac{\pi}{4} \mp \frac{ax}{2} \right) + C.

∫x1+sin⁡axdx=−xatan⁡(π4−ax2)+1a2ln⁡∣1+sin⁡ax∣+C.\int \frac{x}{1 + \sin ax} dx = -\frac{x}{a} \tan \left( \frac{\pi}{4} - \frac{ax}{2} \right) + \frac{1}{a^2} \ln |1 + \sin ax| + C.

∫x1−sin⁡axdx=xatan⁡(π4+ax2)+1a2ln⁡∣1−sin⁡ax∣+C.\int \frac{x}{1 - \sin ax} dx = \frac{x}{a} \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) + \frac{1}{a^2} \ln |1 - \sin ax| + C.

∫sin⁡ax1±sin⁡axdx=±x+1atan⁡(π4∓ax2)+C.\int \frac{\sin ax}{1 \pm \sin ax} dx = \pm x + \frac{1}{a} \tan \left( \frac{\pi}{4} \mp \frac{ax}{2} \right) + C.

∫cos⁡ax1±sin⁡axdx=±1aln⁡∣1±sin⁡ax∣+C.\int \frac{\cos ax}{1 \pm \sin ax} dx = \pm \frac{1}{a} \ln |1 \pm \sin ax| + C.

∫dxsin⁡ax(1±sin⁡ax)=1atan⁡(π4+ax2)+1aln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\sin ax(1\pm\sin ax)} = \frac{1}{a} \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) + \frac{1}{a} \ln \left| \tan \frac{ax}{2} \right| + C.

∫dxsin⁡ax(1±cos⁡ax)=±12a(1±cos⁡ax)+12aln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\sin ax(1\pm\cos ax)} = \pm \frac{1}{2a(1\pm\cos ax)} + \frac{1}{2a} \ln \left| \tan \frac{ax}{2} \right| + C.

∫dxcos⁡ax(1±sin⁡ax)=∓12a(1±sin⁡ax)+12aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{dx}{\cos ax(1\pm\sin ax)} = \mp \frac{1}{2a(1\pm\sin ax)} + \frac{1}{2a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) \right| + C.

∫sin⁡axcos⁡ax(1±sin⁡ax)dx=12a(1±sin⁡ax)±12aln⁡∣tan⁡(π4+ax2)∣+C.\int \frac{\sin ax}{\cos ax(1\pm\sin ax)} dx = \frac{1}{2a(1\pm\sin ax)} \pm \frac{1}{2a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{ax}{2} \right) \right| + C.

∫sin⁡axcos⁡ax(1±cos⁡ax)dx=1aln⁡∣sec⁡ax±1∣+C.\int \frac{\sin ax}{\cos ax(1\pm\cos ax)} dx = \frac{1}{a} \ln |\sec ax \pm 1| + C.

∫cos⁡axsin⁡ax(1±sin⁡ax)dx=−1aln⁡∣csc⁡ax±1∣+C.\int \frac{\cos ax}{\sin ax(1\pm\sin ax)} dx = -\frac{1}{a} \ln |\csc ax \pm 1| + C.

∫cos⁡axsin⁡ax(1±cos⁡ax)dx=−12a(1±cos⁡ax)±12aln⁡∣tan⁡ax2∣+C.\int \frac{\cos ax}{\sin ax(1\pm\cos ax)} dx = -\frac{1}{2a(1\pm\cos ax)} \pm \frac{1}{2a} \ln \left| \tan \frac{ax}{2} \right| + C.

∫dx(1+sin⁡ax)2=−12acot⁡(π4+ax2)−16acot⁡3(π4+ax2)+C.\int \frac{dx}{(1+\sin ax)^2} = -\frac{1}{2a} \cot \left( \frac{\pi}{4} + \frac{ax}{2} \right) - \frac{1}{6a} \cot^3 \left( \frac{\pi}{4} + \frac{ax}{2} \right) + C.

∫dx(1−sin⁡ax)2=−12acot⁡(ax2−π4)−16acot⁡3(ax2−π4)+C.\int \frac{dx}{(1-\sin ax)^2} = -\frac{1}{2a} \cot \left( \frac{ax}{2} - \frac{\pi}{4} \right) - \frac{1}{6a} \cot^3 \left( \frac{ax}{2} - \frac{\pi}{4} \right) + C.

∫sin⁡ax(1+sin⁡ax)2dx=−2a(1tan⁡(ax2+1))+43a(1tan⁡(ax2+1))3+C.\int \frac{\sin ax}{(1+\sin ax)^2} dx = -\frac{2}{a} \left( \frac{1}{\tan \left( \frac{ax}{2} + 1 \right)} \right) + \frac{4}{3a} \left( \frac{1}{\tan \left( \frac{ax}{2} + 1 \right)} \right)^3 + C.

∫sin⁡ax(1−sin⁡ax)2dx=−2a(1tan⁡(ax2−1))+43a(1tan⁡(ax2−1))3+C.\int \frac{\sin ax}{(1-\sin ax)^2} dx = -\frac{2}{a} \left( \frac{1}{\tan \left( \frac{ax}{2} - 1 \right)} \right) + \frac{4}{3a} \left( \frac{1}{\tan \left( \frac{ax}{2} - 1 \right)} \right)^3 + C.

∫dx1+cos⁡ax=−1atan⁡ax2+C.\int \frac{dx}{1+\cos ax} = -\frac{1}{a} \tan \frac{ax}{2} + C.

∫dx1−cos⁡ax=−1acot⁡ax2+C.\int \frac{dx}{1-\cos ax} = -\frac{1}{a} \cot \frac{ax}{2} + C.

∫x1+cos⁡axdx=xatan⁡ax2+2a2ln⁡∣cos⁡ax2∣+C.\int \frac{x}{1+\cos ax} dx = \frac{x}{a} \tan \frac{ax}{2} + \frac{2}{a^2} \ln \left| \cos \frac{ax}{2} \right| + C.

∫x1−cos⁡axdx=−xacot⁡ax2+2a2ln⁡∣sin⁡ax2∣+C.\int \frac{x}{1-\cos ax} dx = -\frac{x}{a} \cot \frac{ax}{2} + \frac{2}{a^2} \ln \left| \sin \frac{ax}{2} \right| + C.

∫sin⁡ax1±cos⁡axdx=∓1aln⁡(1±cos⁡ax)+C.\int \frac{\sin ax}{1\pm\cos ax} dx = \mp \frac{1}{a} \ln (1\pm\cos ax) + C.

∫cos⁡ax1+cos⁡axdx=−1atan⁡ax2+x+C.\int \frac{\cos ax}{1+\cos ax} dx = -\frac{1}{a} \tan \frac{ax}{2} + x + C.

∫cos⁡ax1−cos⁡axdx=−1acot⁡ax2−x+C.\int \frac{\cos ax}{1-\cos ax} dx = -\frac{1}{a} \cot \frac{ax}{2} - x + C.

∫dxcos⁡ax(1+cos⁡ax)=−1atan⁡ax2−ln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\cos ax(1+\cos ax)} = -\frac{1}{a} \tan \frac{ax}{2} - \ln \left| \tan \frac{ax}{2} \right| + C.

∫dxcos⁡ax(1−cos⁡ax)=−1acot⁡ax2−ln⁡∣tan⁡ax2∣+C.\int \frac{dx}{\cos ax(1-\cos ax)} = -\frac{1}{a} \cot \frac{ax}{2} - \ln \left| \tan \frac{ax}{2} \right| + C.

∫dx(1+cos⁡ax)2=12atan⁡ax2+16atan⁡3ax2+C.\int \frac{dx}{(1+\cos ax)^2} = \frac{1}{2a} \tan \frac{ax}{2} + \frac{1}{6a} \tan^3 \frac{ax}{2} + C.

∫dx(1−cos⁡ax)2=−12acot⁡ax2−16acot⁡3ax2+C.\int \frac{dx}{(1-\cos ax)^2} = -\frac{1}{2a} \cot \frac{ax}{2} - \frac{1}{6a} \cot^3 \frac{ax}{2} + C.

∫cos⁡ax(1+cos⁡ax)2dx=12atan⁡ax2−16atan⁡3ax2+C.\int \frac{\cos ax}{(1+\cos ax)^2} dx = \frac{1}{2a} \tan \frac{ax}{2} - \frac{1}{6a} \tan^3 \frac{ax}{2} + C.

∫cos⁡ax(1−cos⁡ax)2dx=12acot⁡ax2−16acot⁡3ax2+C.\int \frac{\cos ax}{(1-\cos ax)^2} dx = \frac{1}{2a} \cot \frac{ax}{2} - \frac{1}{6a} \cot^3 \frac{ax}{2} + C.

∫dx1+cos⁡ax+sin⁡ax=1aln⁡∣1+tan⁡ax2∣+C.\int \frac{dx}{1+\cos ax + \sin ax} = \frac{1}{a} \ln \left| 1 + \tan \frac{ax}{2} \right| + C.

∫dx1+sin⁡2ax=1a2arctan⁡(2tan⁡ax)+C.\int \frac{dx}{1+\sin^2 ax} = \frac{1}{a\sqrt{2}} \arctan (\sqrt{2} \tan ax) + C.

∫dx1+bsin⁡2ax=1a1+barctan⁡(1+btan⁡ax)+C,b>−1.\int \frac{dx}{1+b \sin^2 ax} = \frac{1}{a\sqrt{1+b}} \arctan (\sqrt{1+b} \tan ax) + C , \quad b > -1 .

∫dx1−sin⁡2ax=1atan⁡ax+C.\int \frac{dx}{1-\sin^2 ax} = \frac{1}{a} \tan ax + C.

\frac{1}{a\sqrt{1-b}} \arctan (\sqrt{1-b} \tan ax) + C, & 0 &lt; b &lt; 1, \\ \frac{1}{2a\sqrt{b-1}} \ln \left| \frac{\sqrt{b-1} \tan ax + 1}{\sqrt{b-1} \tan ax - 1} \right| + C, & b > 1. \end{cases} $$ 74. $$ \int \frac{dx}{(1-b \sin^2 ax)^2} = -\frac{b}{2a(1-b)^2} \frac{\tan ax}{\tan^2 ax + \frac{1}{1-b}} + \frac{2-b}{2a\sqrt{1-b}(1-b)} \arctan (\sqrt{1-b} \tan ax) + C , \quad b &lt; 1. $$ 75. $$ \int \frac{\sin ax \cos ax}{1\pm b \sin^2 ax} dx = \pm \frac{1}{2ab} \ln |1 \pm b \sin^2 ax| + C , \quad b \neq 0 . $$ 76. $$ \int \frac{dx}{1+b \cos^2 ax} = \frac{1}{a\sqrt{1+b}} \arctan \frac{\tan ax}{\sqrt{1+b}} + C , \quad b > -1 . $$ 77. $$ \int \frac{dx}{1-\cos^2 ax} = -\frac{1}{a} \cot ax + C. $$ 78. $$ \int \frac{dx}{1-b\cos^2ax} = \left\{ \begin{array}{ll} \frac{1}{a\sqrt{1-b}} \arctan \frac{\tan ax}{\sqrt{1-b}} + C, & 0 &lt; b &lt; 1, \\ \frac{1}{2a\sqrt{b-1}} \ln \left| \frac{\tan ax - \sqrt{b-1}}{\tan ax + \sqrt{b-1}} \right| + C, & b > 1. \end{array} \right. $$ 79. $$ \int \frac{dx}{(1-b\cos^2ax)^2} = \frac{b\sin 2ax}{4a(1-b)(1-b\cos^2ax)} + \frac{2-b}{2a(1-b)} \left\{ \begin{array}{ll} \frac{1}{\sqrt{1-b}} \arctan \frac{\tan ax}{\sqrt{1-b}} + C, & 0 &lt; b &lt; 1, \\ \frac{1}{2\sqrt{b-1}} \ln \left| \frac{\tan ax - \sqrt{b-1}}{\tan ax + \sqrt{b-1}} \right| + C, & b > 1. \end{array} \right. $$ 80. $$ \int \frac{\sin^2ax}{1+b\cos^2ax} dx = -\frac{x}{b} + \frac{\sqrt{1+b}}{ab} \arctan \frac{\tan ax}{\sqrt{1+b}} + C, \quad b > -1, \quad 且 \, b \neq 0. $$ 81. $$ \int \frac{\cos^2ax}{1+b\cos^2ax} dx = \frac{x}{b} - \frac{1}{ab\sqrt{1+b}} \arctan \frac{\tan ax}{\sqrt{1+b}} + C, \quad b > -1, \quad 且 \, b \neq 0. $$ 82. $$ \int \frac{\sin ax \cos ax}{1\pm b\cos^2ax} dx = \pm \frac{1}{ab} \ln \left| \sqrt{1\pm b\cos^2ax} \right| + C, \quad b \neq 0. $$ 83. $$ \int \frac{\sin^2ax}{1-b\cos^2ax} dx = \left\{ \begin{array}{ll} \frac{\sqrt{1-b}}{ab} \arctan \frac{\tan ax}{\sqrt{1-b}} + \frac{x}{b} + C, & 0 &lt; b &lt; 1, \\ \frac{\sqrt{b-1}}{2ab} \ln \left| \frac{\tan ax - \sqrt{b-1}}{\tan ax + \sqrt{b-1}} \right| + \frac{x}{b} + C, & b > 1. \end{array} \right. $$ 84. $$ \int \frac{\cos^2ax}{1-b\cos^2ax} dx = \left\{ \begin{array}{ll} \frac{1}{ab\sqrt{1-b}} \arctan \frac{\tan ax}{\sqrt{1-b}} + \frac{x}{b} + C, & 0 &lt; b &lt; 1, \\ \frac{1}{2ab\sqrt{b-1}} \ln \left| \frac{\tan ax - \sqrt{b-1}}{\tan ax + \sqrt{b-1}} \right| + \frac{x}{b} + C, & b > 1. \end{array} \right. $$ 85. $$ \int \frac{dx}{a+b\sin x} = \left\{ \begin{array}{ll} \frac{2}{\sqrt{a^2-b^2}} \arctan \frac{a\tan \frac{x}{2}+b}{\sqrt{a^2-b^2}} + C, & a^2 > b^2, \\ \frac{1}{\sqrt{b^2-a^2}} \ln \left| \frac{a\tan \frac{x}{2} + b - \sqrt{b^2-a^2}}{a\tan \frac{x}{2} + b + \sqrt{b^2-a^2}} \right| + C, & a^2 &lt; b^2. \end{array} \right. $$ 86. $$ \int \frac{dx}{a+b\cos x} = \left\{ \begin{array}{ll} \frac{2\operatorname{sgn}(a+b)}{\sqrt{a^2-b^2}} \arctan \frac{\sqrt{a^2-b^2}\tan \frac{x}{2}}{|a+b|} + C, & a^2 > b^2, \\ \frac{1}{a} \tan \frac{x}{2} + C, & a = b, \\ -\frac{1}{a} \cot \frac{x}{2} + C, & a = -b, \\ \frac{1}{\sqrt{b^2-a^2}} \ln \left| \frac{\sqrt{b^2-a^2}\tan \frac{x}{2} + |a+b|}{\sqrt{b^2-a^2}\tan \frac{x}{2} - |a+b|} \right| + C, & a^2 &lt; b^2. \end{array} \right. $$ 87. $$ \int \frac{dx}{(a+b\sin x)^2} = \begin{cases} \frac{b\cos x}{(a^2-b^2)(a+b\sin x)} + \frac{2a}{(a^2-b^2)^{3/2}} \arctan \frac{a\tan \frac{x}{2}+b}{\sqrt{a^2-b^2}} + C, & a^2 > b^2, \\ \frac{b\cos x}{(a^2-b^2)(a+b\sin x)} - \frac{a}{(b^2-a^2)^{3/2}} \ln \left| \frac{a\tan \frac{x}{2}+b-\sqrt{b^2-a^2}}{a\tan \frac{x}{2}+b+\sqrt{b^2-a^2}} \right| + C, & a^2 &lt; b^2, \end{cases} $$ 88. $$ \int \frac{dx}{(a+b\cos x)^2} = \frac{b\sin x}{(b^2-a^2)(a+b\cos x)} - \frac{a}{b^2-a^2} \int \frac{dx}{a+b\cos x}. $$ 89. $$ \int \frac{dx}{\sin x(a+b\sin x)} = \frac{1}{a} \ln \left| \tan \frac{x}{2} \right| - \frac{b}{a} \int \frac{dx}{a+b\sin x}. $$ 90. $$ \int \frac{dx}{\cos x(a+b\cos x)} = \frac{1}{a} \ln \left| \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \right| - \frac{b}{a} \int \frac{dx}{a+b\cos x}. $$ 91. $$ \int \frac{\sin x}{a+b\sin x} dx = \frac{x}{b} - \frac{a}{b} \int \frac{dx}{a+b\sin x}. $$ 92. $$ \int \frac{\cos x}{a+b\cos x} dx = \frac{x}{b} - \frac{a}{b} \int \frac{dx}{a+b\cos x}. $$ 93. $$ \int \frac{\cos x}{(a+b\cos x)^2} dx = \frac{a\sin x}{(a^2-b^2)(a+b\cos x)} - \frac{b}{a^2-b^2} \int \frac{dx}{a+b\cos x}. $$ 94. $$ \int \frac{dx}{a^2\cos^2 x + b^2 \sin^2 x} = \frac{1}{ab} \arctan \frac{b \tan x}{a} + C. $$ 95. $$ \int \frac{\sin cx}{a\cos cx + b\sin cx} dx = \frac{1}{c(a^2+b^2)} \left( b c x - a \ln |a \cos cx + b \sin cx| \right) + C. $$ 96. $$ \int \frac{\cos cx}{a\cos cx + b\sin cx} dx = \frac{1}{c(a^2+b^2)} \left( a c x + b \ln |a \cos cx + b \sin cx| \right) + C. $$ 97. $$ \int \frac{\sin cx \cos cx}{a \cos^2 cx + b \sin^2 cx} dx = \frac{1}{2c(b-a)} \ln |a \cos^2 cx + b \sin^2 cx| + C. $$ 98. $$ \int \frac{dx}{a^2 + b^2 \sin^2 cx} = \frac{1}{ac\sqrt{a^2 + b^2}} \arctan \frac{\sqrt{a^2 + b^2} \tan cx}{a} + C. $$ 99. $$ \int \frac{\cos^2 cx}{a^2 + b^2 \sin^2 cx} dx = \frac{\sqrt{a^2 + b^2}}{ab^2 c} \arctan \frac{\sqrt{a^2 + b^2} \tan cx}{a} - \frac{x}{b^2} + C. $$ 100. $$ \int \frac{dx}{a^2 - b^2 \sin^2 cx} = \begin{cases} \frac{1}{ac\sqrt{a^2 - b^2}} \arctan \frac{\sqrt{a^2 - b^2} \tan cx}{a} + C, & a^2 > b^2, \\ \frac{1}{2ac\sqrt{b^2 - a^2}} \ln \left| \frac{\sqrt{b^2 - a^2} \tan cx + a}{\sqrt{b^2 - a^2} \tan cx - a} \right| + C, & a^2 &lt; b^2, \end{cases} $$ 101. $$ \int \frac{dx}{a^2 + b^2 \cos^2 cx} = \frac{1}{ac\sqrt{a^2 + b^2}} \arctan \frac{a \tan cx}{\sqrt{a^2 + b^2}} + C. $$ 102. $$ \int \frac{dx}{a^2 - b^2 \cos^2 cx} = \left\{ \begin{array}{ll} \frac{1}{ac\sqrt{a^2 - b^2}} \arctan \frac{a \tan cx}{\sqrt{a^2 - b^2}} + C, & a^2 > b^2, \\ \frac{1}{2ac\sqrt{b^2 - a^2}} \ln \left| \frac{a \tan cx - \sqrt{b^2 - a^2}}{a \tan cx + \sqrt{b^2 - a^2}} \right| + C, & a^2 &lt; b^2, \end{array} \right. $$ 103. $$ \int \frac{dx}{a^2 + b^2 - 2ab \cos cx} = \frac{2}{c(a^2 - b^2)} \arctan \left( \frac{a+b}{a-b} \tan \frac{cx}{2} \right) + C. $$ 104. $$ \int \frac{x + \sin x}{1 + \cos x} dx = x \tan \frac{x}{2} + C. $$ 105. $$ \int \frac{x - \sin x}{1 - \cos x} dx = -x \cot \frac{x}{2} + C. $$ 106. $$ \int \frac{dx}{\sin ax \pm \cos ax} = \frac{1}{\sqrt{2a}} \ln \left| \tan \left( \frac{ax}{2} \pm \frac{\pi}{8} \right) \right| + C. $$ 107. $$ \int \frac{dx}{(\sin ax \pm \cos ax)^2} = \frac{1}{2a} \tan \left( ax \mp \frac{\pi}{4} \right) + C. $$ 108. $$ \int \frac{\sin ax}{\sin ax \pm \cos ax} dx = \frac{1}{2a} \left( ax \mp \ln |\sin ax \pm \cos ax| \right) + C. $$ 109. $$ \int \frac{\cos ax}{\sin ax \pm \cos ax} dx = \frac{1}{2a} \left( \ln |\sin ax \pm \cos ax| \pm ax \right) + C. $$ 110. $$ \int \frac{dx}{a \cos x + b \sin x} = \frac{\ln \left| \tan \left( \frac{x}{2} + \frac{1}{2} \arctan \frac{a}{b} \right) \right|}{\sqrt{a^2 + b^2}} + C. $$ 111. $$ \int \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x} = \frac{1}{ab} \arctan \left( \frac{b}{a} \tan x \right) + C. $$ 112. $$ \int \frac{dx}{a^2 \cos^2 x - b^2 \sin^2 x} = \frac{1}{2ab} \ln \left| \frac{b \tan x + a}{b \tan x - a} \right| + C. $$ 113. $$ \int x \sin ax dx = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax + C. $$ 114. $$ \int x^2 \sin ax dx = -\frac{1}{a} x^2 \cos ax + \frac{2}{a^2} x \sin ax + \frac{2}{a^3} \cos ax + C. $$ 115. $$ \int x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{1}{a} x \sin ax + C. $$ 116. $$ \int x^2 \cos ax dx = \frac{1}{a} x^2 \sin ax + \frac{2}{a^2} x \cos ax - \frac{2}{a^3} \sin ax + C. $$ *** ## 第四组 含有三角函数的无理式的积分（带根号） 1. $$\int \sqrt{1+\sin ax}\,dx = \mp \frac{2\sqrt{2}}{a}\cos\left(\frac{\pi}{4}+\frac{ax}{2}\right) + C$$ 2. $$\int \sqrt{1-\sin ax}\,dx = \pm \frac{2\sqrt{2}}{a}\sin\left(\frac{\pi}{4}+\frac{ax}{2}\right) + C$$ 3. $$\int \frac{dx}{\sqrt{1+\sin ax}} = \pm \frac{\sqrt{2}}{a}\ln\left|\tan\left(\frac{ax}{4}+\frac{\pi}{8}\right)\right| + C$$ 4. $$\int \frac{dx}{\sqrt{1-\sin ax}} = \pm \frac{\sqrt{2}}{a}\ln\left|\tan\left(\frac{ax}{4}-\frac{\pi}{8}\right)\right| + C$$ *** ## 第五组 含有反三角函数的积分 ($a>0$) 1. $$\int \arcsin\frac{x}{a}\,dx = x\arcsin\frac{x}{a} + \sqrt{a^2-x^2} + C$$ 2. $$\int x\arcsin\frac{x}{a}\,dx = \left(\frac{x^2}{2}-\frac{a^2}{4}\right)\arcsin\frac{x}{a} + \frac{x}{4}\sqrt{a^2-x^2} + C$$ 3. $$\int \arccos\frac{x}{a}\,dx = x\arccos\frac{x}{a} - \sqrt{a^2-x^2} + C$$ 4. $$\int x\arccos\frac{x}{a}\,dx = \left(\frac{x^2}{2}-\frac{a^2}{4}\right)\arccos\frac{x}{a} - \frac{x}{4}\sqrt{a^2-x^2} + C$$ 5. $$\int \arctan\frac{x}{a}\,dx = x\arctan\frac{x}{a} - \frac{a}{2}\ln(a^2+x^2) + C$$ 6. $$\int x\arctan\frac{x}{a}\,dx = \frac{1}{2}(a^2+x^2)\arctan\frac{x}{a} - \frac{a}{2}x + C$$ 7. $$\int \arcsin ax\,dx = x\arcsin ax + \frac{\sqrt{1-a^2x^2}}{a} + C$$ 8. $$\int \arccos ax\,dx = x\arccos ax - \frac{\sqrt{1-a^2x^2}}{a} + C$$ 9. $$\int \arctan ax\,dx = x\arctan ax - \frac{1}{2a}\ln(1+a^2x^2) + C$$ 10. $$\int \text{arccot} ax\,dx = x\text{arccot} ax + \frac{1}{2a}\ln(1+a^2x^2) + C$$ 11. $$\int \text{arcsec} ax\,dx = x\text{arcsec} ax - \frac{1}{a}\ln\left|ax+\sqrt{a^2x^2-1}\right| + C\ (x>0)$$ 12. $$\int \text{arccsc} ax\,dx = x\text{arccsc} ax + \frac{1}{a}\ln\left|ax+\sqrt{a^2x^2-1}\right| + C\ (x>0)$$ 13. $$\int x\arcsin ax\,dx = \frac{1}{4a^2}\left[(2a^2x^2-1)\arcsin ax + ax\sqrt{1-a^2x^2}\right] + C$$ 14. $$\int x\arccos ax\,dx = \frac{1}{4a^2}\left[(2a^2x^2-1)\arccos ax - ax\sqrt{1-a^2x^2}\right] + C$$ 15. $$\int x\arctan ax\,dx = \frac{1+a^2x^2}{2a^2}\arctan ax - \frac{x}{2a} + C$$ 16. $$\int x\text{arccot} ax\,dx = \frac{1+a^2x^2}{2a^2}\text{arccot} ax + \frac{x}{2a} + C$$ 17. $$\int x\text{arcsec} ax\,dx = \frac{x^2}{2}\text{arcsec} ax - \frac{1}{2a^2}\sqrt{a^2x^2-1} + C\ (x>0)$$ 18. $$\int x\text{arccsc} ax\,dx = \frac{x^2}{2}\text{arccsc} ax + \frac{1}{2a^2}\sqrt{a^2x^2-1} + C\ (x>0)$$ 19. $$\int (\arcsin ax)^2\,dx = x(\arcsin ax)^2 - 2x + \frac{2\sqrt{1-a^2x^2}}{a}\arcsin ax + C$$ 20. $$\int (\arccos ax)^2\,dx = x(\arccos ax)^2 - 2x - \frac{2\sqrt{1-a^2x^2}}{a}\arccos ax + C$$ 21. $$\int \frac{\arcsin ax}{x^2}\,dx = -\frac{1}{x}\arcsin ax + a\ln\left|\frac{1-\sqrt{1-a^2x^2}}{ax}\right| + C$$ 22. $$\int \frac{\arccos ax}{x^2}\,dx = -\frac{1}{x}\arccos ax + a\ln\left|\frac{1+\sqrt{1-a^2x^2}}{ax}\right| + C$$ 23. $$\int \frac{\arctan ax}{x^2}\,dx = -\frac{1}{x}\arctan ax - \frac{a}{2}\ln\frac{1+a^2x^2}{a^2x^2} + C$$ 24. $$\int \frac{\text{arccot} ax}{x^2}\,dx = -\frac{1}{x}\text{arccot} ax - \frac{a}{2}\ln\frac{a^2x^2}{1+a^2x^2} + C$$ 25. $$\int \frac{\text{arcsec} ax}{x^2}\,dx = -\frac{1}{x}\text{arcsec} ax + \frac{\sqrt{a^2x^2-1}}{x} + C$$ 26. $$\int \frac{\text{arccsc} ax}{x^2}\,dx = -\frac{1}{x}\text{arccsc} ax - \frac{\sqrt{a^2x^2-1}}{x} + C$$ 27. $$\int \frac{\arcsin ax}{\sqrt{1-a^2x^2}}\,dx = \frac{1}{2a}(\arcsin ax)^2 + C$$ 28. $$\int \frac{\arccos ax}{\sqrt{1-a^2x^2}}\,dx = -\frac{1}{2a}(\arccos ax)^2 + C$$ 29. $$\int \frac{\arctan ax}{1+a^2x^2}\,dx = \frac{1}{2a}(\arctan ax)^2 + C$$ 30. $$\int \frac{\text{arccot} ax}{1+a^2x^2}\,dx = -\frac{1}{2a}(\text{arccot} ax)^2 + C$$ 31. $$\int \arcsin\frac{x}{a}\,dx = x\arcsin\frac{x}{a} + \sqrt{a^2-x^2} + C$$ 32. $$\int \left(\arcsin\frac{x}{a}\right)^2\,dx = x\left(\arcsin\frac{x}{a}\right)^2 + 2\sqrt{a^2-x^2}\arcsin\frac{x}{a} - 2x + C$$ 33. $$\int \left(\arcsin\frac{x}{a}\right)^3\,dx = x\left(\arcsin\frac{x}{a}\right)^3 + 3\sqrt{a^2-x^2}\left(\arcsin\frac{x}{a}\right)^2 - 6x\arcsin\frac{x}{a} - 6\sqrt{a^2-x^2} + C$$ 34. $$\int x\arcsin\frac{x}{a}\,dx = \left(\frac{x^2}{2}-\frac{a^2}{4}\right)\arcsin\frac{x}{a} + \frac{x}{4}\sqrt{a^2-x^2} + C$$ 35. $$\int x^2\arcsin\frac{x}{a}\,dx = \frac{x^3}{3}\arcsin\frac{x}{a} + \frac{x^2+2a^2}{9}\sqrt{a^2-x^2} + C$$ 36. $$\int \frac{x\arcsin x}{\sqrt{1-x^2}}\,dx = x - \sqrt{1-x^2}\arcsin x + C$$ 37. $$\int \frac{x^2\arcsin x}{\sqrt{1-x^2}}\,dx = \frac{x^2}{4} - \frac{x}{2}\sqrt{1-x^2}\arcsin x + \frac{1}{4}(\arcsin x)^2 + C$$ 38. $$\int \frac{x^3\arcsin x}{\sqrt{1-x^2}}\,dx = \frac{\arcsin x\cdot(1-x^2)^{\frac{3}{2}}}{3} + \frac{x^3}{9} - \arcsin x\cdot\sqrt{1-x^2} + \frac{2}{3}x + C$$ 39. $$\int \frac{\arcsin x}{\sqrt{(1-x^2)^3}}\,dx = \frac{x\arcsin x}{\sqrt{1-x^2}} + \frac{1}{2}\ln|1-x^2| + C$$ 40. $$\int \frac{x\arcsin x}{\sqrt{(1-x^2)^3}}\,dx = \frac{\arcsin x}{\sqrt{1-x^2}} + \frac{1}{2}\ln\left|\frac{1-x}{1+x}\right| + C$$ 41. $$\int \arccos\frac{x}{a}\,dx = x\arccos\frac{x}{a} - \sqrt{a^2-x^2} + C$$ 42. $$\int x\arccos\frac{x}{a}\,dx = \frac{x^2}{2}\arccos\frac{x}{a} - \frac{x}{4}\sqrt{a^2-x^2} + \frac{a^2}{4}\arcsin\frac{x}{a} + C$$ 43. $$\int x^2\arccos\frac{x}{a}\,dx = \frac{x^3}{3}\arccos\frac{x}{a} - \frac{x^2+2a^2}{9}\sqrt{a^2-x^2} + C$$ 44. $$\int x\arctan\frac{x}{a}\,dx = \frac{a^2+x^2}{2}\arctan\frac{x}{a} - \frac{ax}{2} + C$$ 45. $$\int x^2\arctan\frac{x}{a}\,dx = \frac{x^3}{3}\arctan\frac{x}{a} + \frac{a^3}{6}\ln(a^2+x^2) - \frac{ax^2}{6} + C$$ 46. $$\int \frac{\arctan\frac{x}{a}}{x^2}\,dx = -\frac{1}{x}\arctan\frac{x}{a} - \frac{1}{a}\ln\left|\frac{a}{x}\sqrt{1+\frac{x^2}{a^2}}\right| + C$$ 47. $$\int \text{arccot}\frac{x}{a}\,dx = x\text{arccot}\frac{x}{a} + \frac{a}{2}\ln(a^2+x^2) + C$$ 48. $$\int x\text{arccot}\frac{x}{a}\,dx = \frac{a^2+x^2}{2}\text{arccot}\frac{x}{a} + \frac{ax}{2} + C$$ 49. $$\int x^2\text{arccot}\frac{x}{a}\,dx = \frac{x^3}{3}\text{arccot}\frac{x}{a} - \frac{a^3}{6}\ln(a^2+x^2) + \frac{ax^2}{6} + C$$ 50. $$\int \frac{\text{arccot}\frac{x}{a}}{x^2}\,dx = -\frac{1}{x}\text{arccot}\frac{x}{a} + \frac{1}{a}\ln\left|\frac{a}{x}\sqrt{1+\frac{x^2}{a^2}}\right| + C$$ *** ## 第六组 含有指数函数的积分 1. $$\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C$$ 2. $$\int xe^{ax}\,dx = \frac{(ax-1)e^{ax}}{a^2} + C$$ 3. $$\int xa^x\,dx = \frac{x}{\ln a}a^x - \frac{a^x}{(\ln a)^2} + C$$ 4. $$\int \frac{dx}{1+e^x} = x - \ln(1+e^x) + C = \ln\frac{e^x}{1+e^x} + C$$ 5. $$\int \frac{dx}{a+be^{px}} = \frac{x}{a} - \frac{1}{ap}\ln(a+be^{px}) + C$$ 6. $$\int \frac{dx}{\sqrt{a+be^{\beta x}}} = \begin{cases} \frac{1}{\beta\sqrt{a}}\ln\frac{\sqrt{a+be^{\beta x}}-\sqrt{a}}{\sqrt{a+be^{\beta x}}+\sqrt{a}} + C, & a>0,b>0, \\ \frac{2}{\beta\sqrt{-a}}\arctan\frac{\sqrt{a+be^{\beta x}}}{\sqrt{-a}} + C, & a&lt;0,b>0. \end{cases}$$ 7. $$\int \frac{dx}{ae^{mx}+be^{-mx}} = \frac{1}{m\sqrt{ab}}\arctan\left(e^{mx}\sqrt{\frac{a}{b}}\right) + C,\ a>0,b>0.$$ 8. $$\int (a^x+a^{-x})\,dx = \frac{a^x+a^{-x}}{\ln a} + C$$ 9. $$\int a^{px}\,dx = \frac{a^{px}}{p\ln a} + C$$ *** ## 第七组 含有对数函数的积分 1. $$\int \ln x\,dx = x\ln x - x + C$$ 2. $$\int \frac{dx}{x\ln x} = \ln|\ln x| + C$$ 3. $$\int x^n\ln x\,dx = \frac{1}{n+1}x^{n+1}\left(\ln x - \frac{1}{n+1}\right) + C$$ 4. $$\int (\ln x)^2\,dx = x(\ln x)^2 - 2x\ln x + 2x + C$$ 5. $$\int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$$ 6. $$\int x^2\ln x\,dx = \frac{x^3}{3}\ln x - \frac{x^3}{9} + C$$ 7. $$\int \frac{\ln x}{(ax+b)^2}\,dx = -\frac{\ln x}{a(ax+b)} + \frac{1}{ab}\ln\left|\frac{x}{ax+b}\right| + C$$ 8. $$\int \ln\frac{x+a}{x-a}\,dx = (x+a)\ln|x+a| - (x-a)\ln|x-a| + C$$ 9. $$\int \frac{1}{x^2}\ln\frac{x+a}{x-a}\,dx = \frac{1}{x}\ln\frac{x-a}{x+a} - \frac{1}{a}\ln\frac{x^2-a^2}{x^2} + C$$ 10. $$\int \ln(x^2+a^2)\,dx = x\ln(x^2+a^2) - 2x + 2a\arctan\frac{x}{a} + C$$ 11. $$\int x\ln(x^2+a^2)\,dx = \frac{1}{2}(x^2+a^2)\ln(x^2+a^2) - \frac{1}{2}x^2 + C$$ 12. $$\int x^2\ln(x^2+a^2)\,dx = \frac{1}{3}\left[x^3\ln(x^2+a^2) - \frac{2}{3}x^3 + 2a^2x - 2a^3\arctan\frac{x}{a}\right] + C$$ 13. $$\int x^{2n}\ln(x^2+a^2)\,dx = \frac{1}{2n+1}\left[x^{2n+1}\ln(x^2+a^2) + (-1)^n2a^{2n+1}\arctan\frac{x}{a} - 2\sum_{k=0}^{n}\frac{(-1)^{n-k}}{2k+1}a^{2n-2k}x^{2k+1}\right] + C$$ 14. $$\int \ln(x^2-a^2)\,dx = x\ln(x^2-a^2) - 2x + a\ln\left|\frac{x+a}{x-a}\right| + C$$ 15. $$\int \ln|x^2-a^2|\,dx = x\ln|x^2-a^2| - 2x + a\ln\left|\frac{x+a}{x-a}\right| + C$$ 16. $$\int x\ln|x^2-a^2|\,dx = \frac{1}{2}\left[(x^2-a^2)\ln|x^2-a^2| - x^2\right] + C$$ 17. $$\int x^2\ln|x^2-a^2|\,dx = \frac{1}{3}\left(x^3\ln|x^2-a^2| - \frac{2}{3}x^3 - 2a^2x + a^3\ln\left|\frac{x+a}{x-a}\right|\right) + C$$ 18. $$\int \sin(\ln x)\,dx = \frac{x}{2}[\sin(\ln x) - \cos(\ln x)] + C$$ 19. $$\int \cos(\ln x)\,dx = \frac{x}{2}[\cos(\ln x) + \sin(\ln x)] + C$$ 20. $$\int x^p\cos(b\ln x)\,dx = \frac{x^{p+1}}{(p+1)^2+b^2}\left[(p+1)\cos(b\ln x) + b\sin(b\ln x)\right] + C$$ 21. $$\int x^p\sin(b\ln x)\,dx = \frac{x^{p+1}}{(p+1)^2+b^2}\left[(p+1)\sin(b\ln x) - b\cos(b\ln x)\right] + C$$
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