Methods from Trigonometric integrals

To evaluate $\int \cos^m x \sin^n x \, dx$ when at least one of $m,n$ is odd:

1. If $m$ is odd, factor out $\cos x$ to obtain $\int \cos^{m-1} x \sin^n x \cos x \, dx$. If $n$ is odd, factor out $\sin x$ to obtain $\int \cos^m x \sin^{n-1} x \sin x \, dx$.
2. The leftover power is even. Replace it using $\sin^2 x+\cos^2 x=1$.
3. Substitute $u=\sin x$ or $u=\cos x$. The binomial expansion $(a+b)^k$ is usually needed before the powers of $u$ can be integrated.

To evaluate $\int \cos^m x \sin^n x \, dx$ when both $m$ and $n$ are even, use

to rewrite the integrand as a sum of integrals of the form $\int \cos^k(2x) \, dx$. Integrate any odd $k$ by the odd-power substitution in $2x$; apply the double-angle identities again to any remaining even $k$.

Convert the product to a sum, then integrate term by term:

To evaluate $\int \tan^m x \sec^n x \, dx$:

• If $n$ is a positive even integer, factor out $\sec^2 x$ and substitute $u=\tan x$. Convert remaining powers of $\sec$ by $1+\tan^2 x=\sec^2 x$.
• If $n=0$, use $\tan^2 x=\sec^2 x-1$ to lower the power of $\tan$ by two, with $\int dx$ and $\int\tan x\,dx$ as base cases.
• If $n$ is odd:
  • If $m$ is odd, factor out $\tan x\sec x$ and substitute $u=\sec x$. Convert remaining powers of $\tan$ by $\tan^2 x=\sec^2 x-1$.
  • If $m$ is even, write $\tan^m x=(\sec^2 x-1)^{m/2}$ and expand into powers of $\sec x$. Reduce odd secant powers by parts until only $\int\sec x\,dx$ remains.