Example $6$

Video Example

Evaluate the following integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cot(x)dx$

Solution

First we write cotangent in terms of sine and cosine.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cot(x)dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{cos(x)}{sin(x)}dx$

This suggests that we should substitute $u=sin(x),$ since then $du=*dx,$ so we get the following. $($Remember to use absolute values where appropriate.$)$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cot(x)dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{cos(x)}{sin(x)}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{du}{u}=ln(|u|)+C$

$=,x+C(in$ terms $ofx)$