# general formulas for integration

1. 1.

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

2. 2.

$\displaystyle\int\lambda\,dx=\lambda x+C$

3. 3.

$\displaystyle\int\lambda f(x)\,dx=\lambda\int f(x)\,dx$

4. 4.

$\displaystyle\int(f(x)+g(x))\,dx=\int f(x)\,dx+\int g(x)\,dx$

5. 5.

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

6. 6.

$\displaystyle\int g(f(x))f^{\prime}(x)\,dx=G(f(x))+C$ββ ifββ $G^{\prime}(t)=g(t)$

7. 7.

$\displaystyle\int[f(x)]^{r}f^{\prime}(x)\,dx=\frac{1}{r\!+\!1}[f(x)]^{r+1}+C$ββ forββ $r\neq-1$

8. 8.

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

9. 9.

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

10. 10.

$\displaystyle\int\!\frac{f(x)}{(f(x)\!+\!a)(f(x)\!+\!b)}\,dx\,=\,\frac{a}{a\!-% \!b}\int\!\frac{dx}{f(x)\!+\!a}-\frac{b}{a\!-\!b}\int\!\frac{dx}{f(x)\!+\!b}$

11. 11.

$\displaystyle\int\sin(\omega x+\varphi)\,dx=-\frac{\cos(\omega x+\varphi)}{% \omega}+C$

12. 12.

$\displaystyle\int\cos(\omega x+\varphi)\,dx=\frac{\sin(\omega x+\varphi)}{% \omega}+C$

13. 13.

$\displaystyle\int\sinh(\omega x+\varphi)\,dx=\frac{\cosh(\omega x+\varphi)}{% \omega}+C$

14. 14.

$\displaystyle\int\cosh(\omega x+\varphi)\,dx=\frac{\sinh(\omega x+\varphi)}{% \omega}+C$

15. 15.

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

16. 16.

$\displaystyle\int\sqrt{ax^{2}\!+\!b}\;dx=\frac{x}{2}\sqrt{ax^{2}\!+\!b}+\frac{% b}{2\sqrt{a}}\ln(x\sqrt{a}+\sqrt{ax^{2}\!+\!b})+C$

17. 17.

$\displaystyle\int\sin^{n}x\cos^{m}x\,dx=-\frac{\sin^{n-1}x\cos^{m+1}x}{m+n}+% \frac{n-1}{m+n}\int\sin^{n-2}x\cos^{m}x\,dx$

18. 18.

$\displaystyle\int\sin^{n}x\cos^{m}x\,dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+% \frac{m-1}{m+n}\int\sin^{n}x\cos^{m-2}x\,dx$

Some series-formed antiderivatives:

$\displaystyle\int f(x)\,dx=C+f(0)x+\frac{f^{\prime}(0)}{2!}x^{2}+\frac{f^{% \prime\prime}(0)}{3!}x^{3}+\ldots$
$\displaystyle\int f(x)\,dx=C+xf(x)-\frac{x^{2}}{2!}f^{\prime}(x)+\frac{x^{3}}{% 3!}f^{\prime\prime}(x)-+\ldots$
$\displaystyle\int UV\,dx=UV^{(-1)}-U^{\prime}V^{(-2)}+U^{\prime\prime}V^{(-3)}% -+\ldots\;=\;\sum_{n=0}^{\infty}(-1)^{n}U^{(n)}V^{(-n-1)}$

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

Title general formulas for integration GeneralFormulasForIntegration 2013-03-22 17:39:31 2013-03-22 17:39:31 pahio (2872) pahio (2872) 16 pahio (2872) Topic msc 26A36 integration formulas TableOfDerivatives IntegralTables IntegrationByParts ReductionFormulasForIntegrationOfPowers