How to evaluate $\int \frac{1+\cos 2x}{\sin^2 2x}$

Question

I'm trying to calculate $\int \frac{1+\cos2x}{\sin^2 2x}$ for the third time and I'm still getting the same result. As usual I tried substitution:

$u = 2x$

$$\frac{1}{2} \int \frac{1+\cos u}{\sin^2 u} = \frac{1}{2} *\int \csc^2 u + \int \frac{\cot u}{\sin u}$$

$$\frac{1}{2} *[-\cot u + \int \cos u ]$$

$$\frac{-\cot 2x}{2} + \frac{\sin 2x}{2}$$

The solution in my textbook is completly different, however. What's wrong with my method?

Answer 1

Again here is the problem $\int \frac{\cot (u)}{ \sin (u) } du= \int \csc( u) \cot( u) du = -\csc (u)$.

So final solution according to your method is :

$$-\frac{1}{2}\left(\cot(2x) + \csc(2x)\right) + c$$

Answer 2

$$\frac{1+\cos2x}{\sin^22x}=\frac{2\cos^2x}{4\sin^2x\cos^2x}=\frac{1}{2\sin^2x}$$ and we got $\cot$.

Answer 3

$${\displaystyle\int}\dfrac{\cos\left(2x\right)+1}{\sin^2\left(2x\right)}\,\mathrm{d}x ={\displaystyle\int}\dfrac{\csc^2\left(x\right)}{2}\,\mathrm{d}x =\class{steps-node}{\cssId{steps-node-1}{\dfrac{1}{2}}}{\displaystyle\int}\csc^2\left(x\right)\,\mathrm{d}x =-\dfrac{\cot\left(x\right)}{2} + C$$