$\int\frac{\sin x}{\cos^2 x}\,dx$ and $\int\csc(ax)\cot(ax)\,dx$

Evaluate \[ \int\frac{\sin x}{\cos^2x}\,dx \qquad\text{and}\qquad \int\csc(ax)\cot(ax)\,dx, \] where $a\ne0$.

Solution

Using $\sin x/\cos^2x=\tan x\sec x$ and $(\sec x)'=\sec x\tan x$, \[ \boxed{\int\frac{\sin x}{\cos^2x}\,dx=\sec x+C.} \] Also $(\csc(ax))'=-a\csc(ax)\cot(ax)$, so \[ \boxed{\int\csc(ax)\cot(ax)\,dx=-\frac1a\csc(ax)+C.} \]