$\int \frac{\sin x}{1+2\sin x}dx$ calculate:
$$\int \frac{\sin x}{1+2\sin x}dx$$
I tried using $\sin x=\dfrac{2u}{u^2+1}$, $u=\tan \dfrac{x}2$ and after Simplification:
$$\int \frac{2u}{u^2+4u+1}×\frac{2}{u^2+1}du$$
and I am not able to calulate that.

Before doing that substitution I might say
$\int \frac{\sin x}{1+2\sin x} \ dx\\
\int \frac 12 \frac{2\sin x + 1 - 1}{1+2\sin x} \ dx\\
\int \frac 12 -  \frac{1}{1+2\sin x} \ dx\\
\int \frac 12 \ dx  -  \frac 12 \int \frac{1}{1+2\sin x} \ dx$
Now when we do the substitution it isn't as messy.
$\frac x2  -   \int \frac{1}{u^2 + 4u + 1} \ du$
