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

• $\displaystyle \int\sin^nx\,dx \;=\; -\frac{1}{n}\sin^{n-1}x\cos{x}+\frac{n\!-\!1}{n}\int\sin^{n-2}x\,dx \qquad (n \gtrless 0)$
• $\displaystyle \int\cos^nx\,dx \;=\; \frac{1}{n}\cos^{n-1}x\sin{x}+\frac{n\!-\!1}{n}\int\cos^{n-2}x\,dx \qquad (n \gtrless 0)$
• $\displaystyle\int(\ln{x})^n\,dx \;=\; x(\ln{x})^n-n\int(\ln{x})^{n-1}\,dx \qquad (n \gtrless 0)$
• $\displaystyle \int\frac{1}{(1+x^2)^n}\,dx \;=\; \frac{1}{2n\!-\!2}\cdot\frac{x}{(1\!+\!x^2)^{n-1}}+\frac{2n\!-\!3}{2n\!-\!2}\int\frac{1}{(1\!+\!x^2)^{n-1}}\,dx \quad (n > 1)$

Example. For finding $\displaystyle\int\!\frac{dx}{\sin^3x}$ , we apply the first formula with $n := -1$ , getting first $$\int\!\frac{dx}{\sin{x}} \,=\, -\frac{1}{-1}\cdot\frac{\cos{x}}{\sin^2x}+\frac{-2}{-1}\int\frac{dx}{\sin^3x}.$$ From this we solve $$\int\!\frac{dx}{\sin^3x} \,=\, -\frac{1}{2}\frac{\cos{x}}{\sin^2x}+\int\!\frac{dx}{\sin{x}} \,=\, -\frac{1}{2}\frac{\cos{x}}{\sin^2x}+\ln\left|\tan\frac{x}{2}\right|+C$$ (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
$\displaystyle\int\sin^{2m+1}x\,dx \,=\, \int\sin^{2m}x\sin{x}\,dx \,=\, -\int(1-\cos^2x)^m(-\sin{x})\,dx$ ,
$\displaystyle\int\cos^{2m+1}x\,dx \,=\, \int\cos^{2m}x\cos{x}\,dx \,=\, \int(1-\sin^2x)^m\cos{x}\,dx,$
$\displaystyle\int\frac{1}{\sin^{2m}x}\,dx \,=\; \int\frac{1}{\sin^{2m-2}x}\cdot\frac{1}{\sin^2x}\,dx \,=\, -\int(1+\cot^2x)^{m-1}\,d\cot{x}$ ,
$\displaystyle\int\frac{1}{\cos^{2m}x}\,dx \,=\; \int\frac{1}{\cos^{2m-2}x}\cdot\frac{1}{\cos^2x}\,dx \,=\, \int(1+\tan^2x)^{m-1}\,d\tan{x}$ ,
which may be found after making the powers on the right hand sides to polynomials.

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