general formulas for integration

1. ${\displaystyle \int f^{\prime }(x)𝑑x}=f(x)+C$

2. ${\displaystyle \int \lambda 𝑑x}=\lambda x+C$

3. ${\displaystyle \int \lambda f(x)𝑑x}=\lambda {\displaystyle \int f(x)𝑑x}$

4. ${\displaystyle \int (f(x)+g(x))𝑑x}={\displaystyle \int f(x)𝑑x}+{\displaystyle \int g(x)𝑑x}$

5. ${\displaystyle \int f(x)g^{\prime }(x)𝑑x}=f(x)g(x)-{\displaystyle \int g(x)f^{\prime }(x)𝑑x}$

6. ${\displaystyle \int g(f(x))f^{\prime }(x)𝑑x}=G(f(x))+C$   if   $G^{\prime }(t)=g(t)$

7. ${\displaystyle \int {[f(x)]}^r{f}^{\prime }(x)𝑑x}={\displaystyle \frac{1}{r+1}}{[f(x)]}^{r+1}+C$   for   $r\ne -1$

8. ${\displaystyle \int \frac{f^{\prime }(x)}{f(x)}𝑑x}=\ln |f(x)|+C$

9. ${\displaystyle \int e^{f(x)}{f}^{\prime }(x)𝑑x}=e^{f(x)}+C$

10. ${\displaystyle \int \frac{f(x)}{(f(x)+a)(f(x)+b)}𝑑x}={\displaystyle \frac{a}{a-b}}{\displaystyle \int \frac{dx}{f(x)+a}}-{\displaystyle \frac{b}{a-b}}{\displaystyle \int \frac{dx}{f(x)+b}}$

11. ${\displaystyle \int \sin (\omega x+\phi )𝑑x}=-{\displaystyle \frac{\cos (\omega x+\phi )}{\omega }}+C$

12. ${\displaystyle \int \cos (\omega x+\phi )𝑑x}={\displaystyle \frac{\sin (\omega x+\phi )}{\omega }}+C$

13. ${\displaystyle \int \sinh (\omega x+\phi )𝑑x}={\displaystyle \frac{\cosh (\omega x+\phi )}{\omega }}+C$

14. ${\displaystyle \int \cosh (\omega x+\phi )𝑑x}={\displaystyle \frac{\sinh (\omega x+\phi )}{\omega }}+C$

15. ${\displaystyle \int \sqrt{ax+b}𝑑x}={\displaystyle \frac{2}{3a}}(ax+b)\sqrt{ax+b}+C$

16. ${\displaystyle \int \sqrt{a{x}^2+b}𝑑x}={\displaystyle \frac{x}{2}}\sqrt{a{x}^2+b}+{\displaystyle \frac{b}{2\sqrt{a}}}\ln (x\sqrt{a}+\sqrt{a{x}^2+b})+C$

17. ${\displaystyle \int {\sin }^{n}x{\cos }^{m}xdx}=-{\displaystyle \frac{{\sin }^{n-1}x{\cos }^{m+1}x}{m+n}}+{\displaystyle \frac{n-1}{m+n}}{\displaystyle \int {\sin }^{n-2}x{\cos }^{m}xdx}$

18. ${\displaystyle \int {\sin }^{n}x{\cos }^{m}xdx}={\displaystyle \frac{{\sin }^{n+1}x{\cos }^{m-1}x}{m+n}}+{\displaystyle \frac{m-1}{m+n}}{\displaystyle \int {\sin }^{n}x{\cos }^{m-2}xdx}$
    

Some series-formed antiderivatives:

${\displaystyle \int f(x)𝑑x}=C+f(0)x+{\displaystyle \frac{f^{\prime }(0)}{2!}}{x}^2+{\displaystyle \frac{f^{\prime \prime }(0)}{3!}}{x}^3+\mathrm{\dots }$
${\displaystyle \int }f(x)dx=C+xf(x)-{\displaystyle \frac{x^2}{2!}}{f}^{\prime }(x)+{\displaystyle \frac{x^3}{3!}}{f}^{\prime \prime }(x)-+\mathrm{\dots }$
${\displaystyle \int }UVdx=U{V}^{(-1)}-U^{\prime }{V}^{(-2)}+U^{\prime \prime }{V}^{(-3)}-+\mathrm{\dots }={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{(-1)}^n{U}^{(n)}{V}^{(-n-1)}$

The derivatives with negative order (http://planetmath.org/HigherOrderDerivatives) that $V$ has been integrated repeatedly.