reduction formulas for integration of powers

The following reduction formulas, with integer $n$ and via integration by parts, may be used for lowing ($n>0$) or raising () the the powers:

• ${\displaystyle \int }{\sin }^{n}xdx=-{\displaystyle \frac{1}{n}}{\sin }^{n-1}x\cos x+{\displaystyle \frac{n-1}{n}}{\displaystyle \int }{\sin }^{n-2}xdx\mathit{\hspace{1em}\hspace{1em}}(n\gtrless 0)$

• ${\displaystyle \int }{\cos }^{n}xdx={\displaystyle \frac{1}{n}}{\cos }^{n-1}x\sin x+{\displaystyle \frac{n-1}{n}}{\displaystyle \int }{\cos }^{n-2}xdx\mathit{\hspace{1em}\hspace{1em}}(n\gtrless 0)$

• ${\displaystyle \int }{(\ln x)}^{n}dx=x{(\ln x)}^n-n{\displaystyle \int }{(\ln x)}^{n-1}dx\mathit{\hspace{1em}\hspace{1em}}(n\gtrless 0)$

• ${\displaystyle \int }{\displaystyle \frac{1}{{(1+x^2)}^n}}dx={\displaystyle \frac{1}{2n-2}}\cdot {\displaystyle \frac{x}{{(1+x^2)}^{n-1}}}+{\displaystyle \frac{2n-3}{2n-2}}{\displaystyle \int }{\displaystyle \frac{1}{{(1+x^2)}^{n-1}}}dx\mathit{\hspace{1em}}(n>1)$

Example.  For finding ${\displaystyle \int \frac{dx}{{\sin }^{3}x}}$, we apply the first formula with  $n:=-1$,  getting first

$$\int \frac{dx}{\sin x}=-\frac{1}{-1}\cdot \frac{\cos x}{{\sin }^{2}x}+\frac{-2}{-1}\int \frac{dx}{{\sin }^{3}x}.$$

From this we solve

$$\int \frac{dx}{{\sin }^{3}x}=-\frac{1}{2}\frac{\cos x}{{\sin }^{2}x}+\int \frac{dx}{\sin x}=-\frac{1}{2}\frac{\cos x}{{\sin }^{2}x}+\ln \left|\tan \frac{x}{2}\right|+C$$

(see integration of rational function of sine and cosine).

Note 1.  Instead of the two first formulae, it is simpler in the cases when $n$ is a positive odd or a negative even number to use the following
${\displaystyle \int {\sin }^{2m+1}xdx}={\displaystyle \int {\sin }^{2m}x\sin xdx}=-{\displaystyle \int {(1-{\cos }^{2}x)}^m(-\sin x)𝑑x}$,
${\displaystyle \int {\cos }^{2m+1}xdx}={\displaystyle \int {\cos }^{2m}x\cos xdx}={\displaystyle \int {(1-{\sin }^{2}x)}^m\cos xdx},$
${\displaystyle \int \frac{1}{{\sin }^{2m}x}𝑑x}={\displaystyle \int \frac{1}{{\sin }^{2m-2}x}\cdot \frac{1}{{\sin }^{2}x}𝑑x}=-{\displaystyle \int {(1+{\cot }^{2}x)}^{m-1}d\cot x}$,
${\displaystyle \int \frac{1}{{\cos }^{2m}x}𝑑x}={\displaystyle \int \frac{1}{{\cos }^{2m-2}x}\cdot \frac{1}{{\cos }^{2}x}𝑑x}={\displaystyle \int {(1+{\tan }^{2}x)}^{m-1}d\tan x}$,
which may be found after making the powers on the right hand sides to polynomials.

Note 2.  $\int {\tan }^{n}xdx$  ($n\in {\mathbb{Z}}_{+}$)  is obtained easily by the substitution (http://planetmath.org/IntegrationBySubstitution)  $\tan x:=t$,  $dx=\frac{dt}{t^2+1}$  and a division; e.g.

${\displaystyle \int {\tan }^{5}xdx}$ $\mathrm{\hspace{0.17em}}={\displaystyle \int \frac{t^5}{t^2+1}𝑑t}={\displaystyle \int \left(t^3-t+\frac{t}{t^2+1}\right)𝑑t}$

$\mathrm{\hspace{0.17em}}={\displaystyle \frac{t^4}{4}}-{\displaystyle \frac{t^2}{2}}+{\displaystyle \frac{1}{2}}\ln (t^2+1)+C$

$\mathrm{\hspace{0.17em}}={\displaystyle \frac{{\tan }^{4}x}{4}}-{\displaystyle \frac{{\tan }^{2}x}{2}}+\ln \sqrt{{\tan }^{2}x+1}+C.$

Title	reduction formulas for integration of powers
Canonical name	ReductionFormulasForIntegrationOfPowers
Date of creation	2013-03-22 18:36:53
Last modified on	2013-03-22 18:36:53
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Topic
Classification	msc 26A36
Classification	msc 26A09
Synonym	integration of powers
Related topic	GeneralFormulasForIntegration
Related topic	IntegralTables
Related topic	WallisFormulae