Question

Integration of
$\int \frac{x^2}{1+y^2}dx$

Answer 1

Given that,

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

$\begin{array}{c}\int a.f\left(x\right)dx=a.\int f\left(x\right)dx\\ =\frac{1}{1+y^2}.\int x^{2}dx\end{array}$

We know that,
$\int x^{a}dx=\frac{x^{a+1}}{a+1}$

Therefore,

$I=\frac{1}{1+y^2}.\frac{x^2+1}{2+1}+C$

$I=\frac{x^3}{3\left(1+y^2\right)}+C$

Hence, this is the answer.