Integral rules of Trigonometric functions

In integral calculus, the trigonometric functions are involved in integration but the integrals of trigonometric functions cannot be evaluated directly and it requires some special rules to find the integrals of them. The following are some standard trigonometric integration formulas with proofs.

Basic Rules

The list of fundamental integral rules of trigonometric functions with proofs.

Sine

$\displaystyle \int{\sin{x}\,}dx$ $\,=\,$ $-\cos{x}+c$

Cosine

$\displaystyle \int{\cos{x}\,}dx$ $\,=\,$ $\sin{x}+c$

Square of secant

$\displaystyle \int{\sec^2{x}\,}dx$ $\,=\,$ $\tan{x}+c$

Square of cosecant

$\displaystyle \int{\csc^2{x}\,}dx$ $\,=\,$ $-\cot{x}+c$

Product of Secant and Tangent

$\displaystyle \int{\sec{x}\tan{x}\,}dx$ $\,=\,$ $\sec{x}+c$

Product of Cosecant and Cotangent

$\displaystyle \int{\csc{x}\cot{x}\,}dx$ $\,=\,$ $-\csc{x}+c$

$\displaystyle \int{\operatorname{cosec}{x}\cot{x}\,}dx$ $\,=\,$ $-\operatorname{cosec}{x}+c$

Advanced Rules

The list of advanced trigonometric integration formulas with proofs.

Secant

$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\log_{e}{|\sec{x}+\tan{x}|}+c$

Cosecant

$\displaystyle \int{\csc{x}}\,dx$ $\,=\,$ $\log_{e}{|\csc{x}-\cot{x}|}+c$

$\displaystyle \int{\operatorname{cosec}{x}}\,dx$ $\,=\,$ $\log_{e}{|\operatorname{cosec}{x}-\cot{x}|}+c$

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