# How do you integrate int tcsctcott by parts?

Feb 18, 2017

$\int t \csc t \cot t \mathrm{dt} = - t \csc t - \ln | \csc t + \cot t | + C$

#### Explanation:

Let $\mathrm{dv} = \csc t \cot t \mathrm{dt}$ and $u = t$. This means that $v = - \csc t$ and $\mathrm{du} = \mathrm{dt}$.

Through integration by parts, we have:

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

$\int t \csc t \cot t \mathrm{dt} = t \left(- \csc t\right) - \int \left(- \csc t \mathrm{dt}\right)$

$\int t \csc t \cot t \mathrm{dt} = - t \csc t + \int \csc t \mathrm{dt}$

The integral of $\csc t$ can be found here

$\int t \csc t \cot t \mathrm{dt} = - t \csc t - \ln | \csc t + \cot t | + C$

Hopefully this helps!