積分公式表

1. $$\int x^n\, dx= \frac{1}{n+1} x^{n+1}+C.$$
2. $$\int \frac{1}{x}\, dx= \ln \vert x\vert+C.$$
3. $$\int e^x\, dx=e^x+C.$$
4. $$\int \sin x\, dx=- \cos x +C.$$
5. $$\int \cos x\, dx= \sin x +C.$$
6. $$\int \sec^2 x\, dx= \tan x +C.$$
7. $$\int \csc^2 x\, dx=- \cot x +C.$$
8. $$\int \sec x \tan x\, dx= \sec x +C.$$
9. $$\int \csc x \cot x\, dx= -\csc x +C.$$
10. $$\int \tan x\, dx= \ln \vert \sec x \vert +C.$$
11. $$\int \cot x\, dx= \ln \vert \sin x \vert +C.$$
12. $$\int \sec x\, dx= \ln \vert \sec x + \tan x \vert +C.$$
13. $$\int \csc x\, dx=- \ln \vert \csc x + \cot x \vert +C.$$
14. $$\int \frac{dx}{\sqrt{1-x^2}} = \arcsin x +C.$$
15. $$\int \frac{dx}{\sqrt{1+x^2}} = \ln \vert x+ \sqrt{1+x^2} \vert+C.$$
16. $$\int \frac{dx}{\sqrt{x^2-1}} = \ln \vert x+ \sqrt{x^2-1} \vert+C.$$
17. $$\int \frac{dx}{1+x^2} = \arctan x +C.$$
18. $$\int \frac{dx}{1-x^2} = \frac{1}{2} \ln \left \vert \frac{1+x}{1-x} \right \vert+C.$$
19. $$\int \sqrt{1-x^2}\, dx = \frac{1}{2} (x \sqrt{1-x^2} + \arcsin x)+C.$$
20. $$\int \sqrt{1+x^2}\, dx = \frac{1}{2} [x \sqrt{1+x^2} + \ln (x+ \sqrt{1+x^2} )]+C.$$
21. $$\int \sqrt{x^2-1}\, dx = \frac{1}{2} (x \sqrt{x^2-1} - \ln \vert x+ \sqrt{x^2-1} \vert)+C.$$
22. $$\int \sin^{n}x\, dx= \left\{\begin{array}{cc} \displaystyle-\frac{... ...\geq2, \\ \noalign{\smallskip } -\textup{cos}x+c, & n=1. \end{array} \right.$$
23. $$\int \textup{cos}^{n}x\, dx= \left\{\begin{array}{cc} \displaysty... ...n\geq2, \\ \noalign{\smallskip } \textup{sin}x+c, & n=1. \end{array} \right.$$
24. $$\int \textup{sin}^{n}x\textup{cos}^{m}x\, dx =-\frac{\textup{sin... ...os}^{m+1}x}{n+m} +\frac{n-1}{m+n}\int \textup{sin}^{n-2}x\textup{cos}^{m}x\, dx$$ $$=\;\frac{\textup{sin}^{n+1}x\textup{cos}^{m-1}x}{n+m} +\frac{m-1}{m+n}\int \textup{sin}^{n}x\textup{cos}^{m-2}x\, dx.$$
25. $$\int \frac{dx}{[(x+\alpha)^2+\beta^2]^n} =\frac{x+\alpha}{2(n-1)\beta^2[(x+\alpha)^2+\beta^2]^{n-1}}$$ $$+\;\frac{2n-3}{2(n-1)\beta^2}\cdot\int \frac{dx}{[(x+\alpha)^2+\beta^2]^{n-1}}$$