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.

Title	general formulas for integration
Canonical name	GeneralFormulasForIntegration
Date of creation	2013-03-22 17:39:31
Last modified on	2013-03-22 17:39:31
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Topic
Classification	msc 26A36
Synonym	integration formulas
Related topic	TableOfDerivatives
Related topic	IntegralTables
Related topic	IntegrationByParts
Related topic	ReductionFormulasForIntegrationOfPowers