Chapter 7 - Section 7.1 - The Logarithm Defined as an Integral - Exercises - Page 401: 22

$-e^{\csc(\pi+t)} +c$

Solve $\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt$ Let us $p=\csc(\pi+t)$ and $dp=-\csc(\pi+t) \cot(\pi+t)dt$ This implies $\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt=-\int e^p dp$ $=- e^p +c$ $=-e^{\csc(\pi+t)} +c$ Hence,$\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt=-e^{\csc(\pi+t)} +c$