# Integration by Substitution/Examples/Cosine over Square of 1 + Sine

## Example of Use of Integration by Substitution

$\ds \int \dfrac {\cos x} {\paren {1 + \sin x}^2} \rd x = -\dfrac 1 {1 + \sin x} + C$

## Proof

 $\ds u$ $=$ $\ds \sin x$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds \cos x$ Power Rule for Derivatives $\ds \leadsto \ \$ $\ds \int \dfrac {\cos x} {\paren {1 + \sin x}^2} \rd x$ $=$ $\ds \int \dfrac 1 {\paren {1 + u}^2} \rd u$ Primitive of Composite Function: Corollary $\ds$ $=$ $\ds -\dfrac 1 {1 + u} + C$ Primitive of Power $\ds$ $=$ $\ds -\dfrac 1 {1 + \sin x} + C$ substituting for $u$ and simplifying

$\blacksquare$