Question

Evaluate the following indefinite integrals. Be sure to clearly identify your u and du.

5. ſcot (90) do

Answer 1

5)

$\small \int cot(9x)dx$

Substitute $\small u=9x\rightarrow du=9dx$

$\small \Rightarrow dx=\frac{du}{9}$

The integration becomes  

du cot(u) 9

I cot (uldu 9

We need to find

$\small \int cot(u)du$

$\small =\int \frac{cos(u)}{sin(u)}du$

Substitute $\small v=sin(u)\rightarrow dv=cos(u)du$

So, integration becomes

$\small \int \frac{1}{v}dv=ln|v|+c$

$\small =ln|sin(u)|+c$   $\small (\because v=sin(u))$

Putting the value in the original integral, we get

$\small \frac{1}{9}\int cot(u)du=\frac{1}{9}ln|sin(u)|+c$

$\small =\frac{1}{9}ln|sin(9x)|+c$   $\small (\because u=9x)$

$\small \therefore \int cot(9x)dx=\frac{1}{9}ln|sin(9x)|+C$, where C is a constant.