Question

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

Answer 1

Consider the given integral.

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

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

$I=\int \frac{x(1+2x^2)}{(x^2+x^4)}dx$

Let $t=x^2+x^4$

$\frac{dt}{dx}=2x+4x^3$

$\frac{dt}{2}=x(1+2x^2)$

Therefore,

$I=\frac{1}{2}\int \frac{1}{t}dt$

$I=\frac{1}{2}\ln \left(t\right)+C$

On putting the value of $t$, we get

$I=\frac{1}{2}\ln (x^2+x^4)+C$

Hence, this is the answer.