Question

question 17
evaluate: $\int \csc \pi x \cot \pi x dx$.
no correct answer choice is given.
$-\frac{1}{\pi} \cot(\pi x)+c$
$\pi \csc(\pi x)+c$
$-\pi \cot(\pi x)+c$
$-\frac{1}{\pi} \csc(\pi x)+c$

Explanation:

Step1: Use substitution

Let \(u = \pi x\), then \(du=\pi dx\), and \(dx=\frac{1}{\pi}du\).
The integral \(\int\csc(\pi x)\cot(\pi x)dx\) becomes \(\frac{1}{\pi}\int\csc(u)\cot(u)du\).

Step2: Recall the integral formula

We know that \(\int\csc(u)\cot(u)du=-\csc(u)+C\).
Substituting back \(u = \pi x\), we get \(\frac{1}{\pi}(-\csc(u))+C=-\frac{1}{\pi}\csc(\pi x)+C\).

Answer:

\(-\frac{1}{\pi}\csc(\pi x)+C\) (the last option)