# 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 ($n<0$) the the powers:

• $\displaystyle\int\sin^{n}x\,dx\;=\;-\frac{1}{n}\sin^{n-1}x\cos{x}+\frac{n\!-\!% 1}{n}\int\sin^{n-2}x\,dx\qquad(n\gtrless 0)$

• $\displaystyle\int\cos^{n}x\,dx\;=\;\frac{1}{n}\cos^{n-1}x\sin{x}+\frac{n\!-\!1% }{n}\int\cos^{n-2}x\,dx\qquad(n\gtrless 0)$

• $\displaystyle\int(\ln{x})^{n}\,dx\;=\;x(\ln{x})^{n}-n\int(\ln{x})^{n-1}\,dx% \qquad(n\gtrless 0)$

• $\displaystyle\int\frac{1}{(1+x^{2})^{n}}\,dx\;=\;\frac{1}{2n\!-\!2}\cdot\frac{% x}{(1\!+\!x^{2})^{n-1}}+\frac{2n\!-\!3}{2n\!-\!2}\int\frac{1}{(1\!+\!x^{2})^{n% -1}}\,dx\quad(n>1)$

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

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

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
$\displaystyle\int\sin^{2m+1}x\,dx\,=\,\int\sin^{2m}x\sin{x}\,dx\,=\,-\int(1-% \cos^{2}x)^{m}(-\sin{x})\,dx$,
$\displaystyle\int\cos^{2m+1}x\,dx\,=\,\int\cos^{2m}x\cos{x}\,dx\,=\,\int(1-% \sin^{2}x)^{m}\cos{x}\,dx,$
$\displaystyle\int\frac{1}{\sin^{2m}x}\,dx\,=\;\int\frac{1}{\sin^{2m-2}x}\cdot% \frac{1}{\sin^{2}x}\,dx\,=\,-\int(1+\cot^{2}x)^{m-1}\,d\cot{x}$,
$\displaystyle\int\frac{1}{\cos^{2m}x}\,dx\,=\;\int\frac{1}{\cos^{2m-2}x}\cdot% \frac{1}{\cos^{2}x}\,dx\,=\,\int(1+\tan^{2}x)^{m-1}\,d\tan{x}$,
which may be found after making the powers on the right hand sides to polynomials.

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

 $\displaystyle\int\tan^{5}x\,dx$ $\displaystyle\,=\,\int\frac{t^{5}}{t^{2}\!+\!1}\,dt\,=\,\int\!\left(t^{3}-t+% \frac{t}{t^{2}\!+\!1}\right)dt$ $\displaystyle\,=\,\frac{t^{4}}{4}-\frac{t^{2}}{2}+\frac{1}{2}\ln(t^{2}\!+\!1)+C$ $\displaystyle\,=\,\frac{\tan^{4}x}{4}-\frac{\tan^{2}x}{2}+\ln\sqrt{\tan^{2}x+1% }+C.$
Title reduction formulas for integration of powers ReductionFormulasForIntegrationOfPowers 2013-03-22 18:36:53 2013-03-22 18:36:53 pahio (2872) pahio (2872) 10 pahio (2872) Topic msc 26A36 msc 26A09 integration of powers GeneralFormulasForIntegration IntegralTables WallisFormulae