Math定积分简化

定积分简化

  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. ∫x1+x4dx=12arctan⁡x2+C\int \frac{x}{1+x^{4}} d x=\frac{1}{2} \arctan x^{2}+C
  11. ∫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
  12. ∫dxc2+x2=1carctan⁡xc+C\int \frac{d x}{c^{2}+x^{2}}=\frac{1}{c} \arctan \frac{x}{c}+C
  13. ∫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
  14. ∫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
  15. ∫x2c2+x2dx=x−carctan⁡xc+C\int \frac{x^{2}}{c^{2}+x^{2}} d x=x-c \arctan \frac{x}{c}+C
  16. ∫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
  17. ∫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
  18. ∫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
  19. ∫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
  20. ∫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
  21. ∫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
  22. ∫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
  23. ∫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
  24. ∫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
  25. ∫ax+bdx=23a(ax+b)3+C\int \sqrt{a x+b} d x=\frac{2}{3 a} \sqrt{(a x+b)^{3}}+C
  26. ∫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}
  27. ∫xx2−a2dx=x2−a2+C\int \frac{x}{\sqrt{x^{2}-a^{2}}} d x=\sqrt{x^{2}-a^{2}}+C
  28. ∫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
  29. ∫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
  30. ∫dxxx2−a2=1aarccos⁡a∣x∣+C\int \frac{d x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \arccos \frac{a}{|x|}+C
  31. ∫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
  32. ∫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
  33. ∫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
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