Primitives of Trigonometric Functions

Theorem

This page gathers together primitives of trigonometric functions.

In the below, $C$ is an arbitrary constant throughout.

Primitive of Sine Function

$\ds \int \sin x \rd x = -\cos x + C$

Primitive of Cosine Function

$\ds \int \cos x \rd x = \sin x + C$

Primitive of Tangent Function: Cosine Form

$\ds \int \tan x \rd x = -\ln \size {\cos x} + C$

where $\cos x \ne 0$.

Primitive of Tangent Function: Secant Form

$\ds \int \tan x \rd x = \ln \size {\sec x} + C$

where $\sec x$ is defined.

Primitive of Cotangent Function

$\ds \int \cot x \rd x = \ln \size {\sin x} + C$

where $\sin x \ne 0$.

Primitive of Secant Function: Secant plus Tangent Form

$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$

where $\sec x + \tan x \ne 0$.

Primitive of Secant Function: Tangent plus Angle Form

$\ds \int \sec x \rd x = \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C$

Primitive of Cosecant Function: Cosecant plus Cotangent Form

$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$

where $\csc x + \cot x \ne 0$.

Primitive of Cosecant Function: Cosecant minus Cotangent Form

$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$

where $\csc x - \cot x \ne 0$.

Primitive of Cosecant Function: Tangent Form

$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$

where $\tan \dfrac x 2 \ne 0$.