Question

find the indefinite integral of csc(5x)

Answer 1

u = 5x
du = 5 dx
\[\int\limits \csc(5x) dx = \frac{1}{5}\int\limits \csc (u) du\]
trick to this integral is multiply top/bottom by csc - cot
\[\rightarrow \frac{1}{5} \int\limits \frac{\csc^{2} u - \csc u \cot u}{\csc u - \cot u} du\]
w = csc - cot
dw = csc^2 - csc*cot
\[\rightarrow \frac{1}{5} \int\limits \frac{dw}{w} = \frac{1}{5}\ln w+C\]
\[= \frac{1}{5} \ln |\csc(5x) -\cot(5x)| + C\]