Question

Integrate the following functions.
$\int \frac{2x}{1+x^2}dx.$

Answer 1

Let $I=\int \frac{2x}{1+x^2}dx$
Let $1+x^2=t$
On differentiating w.r.t.x, we get 2x dx $=dt\Rightarrow dx=\frac{dt}{2x}$
$\therefore I=\int \frac{2x}{t}\frac{dt}{2x}=\int \frac{1}{t}dt=log|t|+C=log|1+x^2|+C$