How to integrate trigonometric functions

Some trig integrals work by recognizing a derivative we already know. Others give us a logarithm. Let’s use the standard antiderivatives and see how an inside factor changes the result.

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Use the standard antiderivatives

Trig antiderivatives with input u

Integrand	Antiderivative, plus C
$\sin u$	$-\cos u$
$\cos u$	$\sin u$
$\tan u$	$-\ln \left|\cos u\right|$
$\cot u$	$\ln \left|\sin u\right|$
$\sec u$	$\ln |\sec u+\tan u|$
$\csc u$	$-\ln |\csc u+\cot u|$
${\sec }^{2}u$	$\tan u$
${\csc }^{2}u$	$-\cot u$
$\sec u\tan u$	$\sec u$
$\csc u\cot u$	$-\csc u$

If the input is $u=ax$ with $a\ne 0$, multiply the antiderivative by $\frac{1}{a}$. This is the factor from $du=adx$. Use each formula on an interval where the integrand is defined.

Integrate tangent with a factor inside

$\int 6\tan \left(2x\right)dx=-3\ln \left|\cos \right(2x\left)\right|+C$

Integrate a sum of different functions

$\int (e^{4x}-\sec (7x\left)\tan \right(7x)+5\sin x)dx$

$=\frac{e^{4x}}{4}-\frac{\sec \left(7x\right)}{7}-5\cos x+C$

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