Integrals of Trig Functions

Key Trigonometric Identities

• Make good use of ${\displaystyle 1}$:
  • ${\displaystyle {\sin }^{2}x+{\cos }^{2}x=1}$
  • ${\displaystyle {\tan }^{2}x={\sec }^{2}x-1}$, with the identity below can be derived by the equation above.
  • ${\displaystyle {\cot }^{2}x={\csc }^{2}x-1}$
  • ${\displaystyle 2{\cos }^{2}x-1=\cos 2x}$, which is the half-angle identity.

Functions With ${\displaystyle \sin x}$ and ${\displaystyle \cos x}$

${\displaystyle \int {\sin }^{n}xdx,\int con{s}^{n}xdx}$

When ${\displaystyle n}$ in odd, use ${\displaystyle {\sin }^{2}x+{\cos }^{2}x=1}$ to substitute some part of the original function with only one ${\displaystyle \sin x}$ or ${\displaystyle \cos x}$ left, then use the basic substitution.

When ${\displaystyle n}$ in even, use ${\displaystyle 2{\cos }^{2}x-1=\cos 2x}$ to substitute the integrand.

${\displaystyle \int {\sin }^{m}x{\cos }^{n}xdx}$

When ${\displaystyle m}$ or ${\displaystyle n}$ in odd, use ${\displaystyle {\sin }^{2}x+{\cos }^{2}x=1}$ to substitute until only one ${\displaystyle \sin x}$ or ${\displaystyle \cos x}$ left, then use the basic substitution.

When both ${\displaystyle m}$ and ${\displaystyle n}$ in even, use the half-angle identity to substitute all ${\displaystyle \sin x}$ and ${\displaystyle \cos x}$ into ${\displaystyle \sin x}$ or ${\displaystyle \cos x}$, then expand the formula, integrate them one by one.

${\displaystyle \int \sin mx\cos nxdx}$

Use the identity ${\displaystyle \sin (m+n)x=\sin mx\cos nx+\cos mx\sin nx}$ to make the product into addition or subtraction.

Functions With ${\displaystyle \tan x}$ and ${\displaystyle \sec x}$

Some preliminaries are required:

• ${\displaystyle \int \tan xdx=\ln |\sec x|=-\ln |\cos x|}$
• ${\displaystyle \int \sec xdx=\ln |\tan x+\sec x|}$
• ${\displaystyle d\tan x={\sec }^{2}xdx}$
• ${\displaystyle d\sec x=\sec x\tan xdx}$

With these identities, many functions with ${\displaystyle \tan x}$ and ${\displaystyle \sec x}$ can be handled.