Integral of $\sin (2x)/(1+\cos^2 x)$

Question

The question is to find

$$\displaystyle \int \frac {\sin (2x)}{1+\cos^2x}.$$

Can anyone help me? I need all the steps, because I need to understand what to do. Thank you.

Answer 1

You can use the identity: $$ \sin(2x)=2\sin(x)\cos(x). $$ Then use a $u$-substitution with $u=1+\cos^2(x)$.

Answer 2

Note that $\sin(2x)=2\sin(x)\cos(x)$

Therefore the problem reduces to finding the integral: $\int \frac {2\sin(x)\cos(x)}{1+\cos^2(x)}dx=-\log(1+\cos^2(x))+C$

Answer 3

By the double angle formula, $\sin(2x) = 2 \sin(x)\cos(x)$

$$\int \frac{\sin(2x)}{1 + \cos^2(x)}dx = \int \frac{2\sin(x)\cos(x)}{1+\cos^2(x)}dx$$

Let $u = 1 + \cos^2(x)$ $du = -2\sin(x)\cos(x) dx$

so...substituting, we get: $$\int \frac{2\sin(x)\cos(x)}{1+\cos^2(x)}dx = \int -\frac{1}{u} du$$

Can you take it from here?

Integrate with respect to $u$, then "back" substitute $u = 1 + \cos^2(x)$ into the result.