Question

Integrate: cot^2(x+pi/4)
(See inside)

Answer 1

\[\int\limits \cot^{2}(x+\frac{\pi}{4})\]
I know that the answer is eventually something close to\[-\csc(x+\frac{\pi}{4})\]

Answer 2

But the answer key gives
\[-\csc(x+\frac{\pi}{4})-x \]
So what part of the chain rule am I forgetting?

Answer 3

\[\large \color{salmon}{\cot^2x+1=\csc^2x} \qquad \rightarrow \qquad \color{salmon}{\cot^2x=\csc^2x-1}\]Using this pink identity gives us,\[\large \int\limits \cot^2(x+\frac{\pi}{4})dx \qquad \rightarrow \qquad \int\limits \csc^2(x+\frac{\pi}{4})-1\quad dx\]

Answer 4

This shouldn't be too difficult to integrate.
I know cotangents and cosecants don't come up as often as the others but this is still a good one to remember :)

Answer 5

\[\large \int\limits \sec^2x dx = \tan x, \qquad \qquad \int\limits \csc^2x dx=-\cot x\]

Answer 6

Thanks. Didn't realize it was an identity.