Question

Solve:
$\int \frac{3x^2}{x^6+1}dx$

Answer 1

$\int \frac{3x^2}{x^6+1}\cdot dx$

$\int \frac{3x^2}{(x^3{)}^2+1}\cdot dx$

Let $x^3=t$

Differentiate $3x^2\cdot dx=dt$

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

$[\because \frac{dx}{x^2+a^2}=\frac{1}{a}ta{n}^{-1}\frac{x}{a}+C]$

$\int \frac{3x^2}{t^2+1}\cdot \frac{dt}{3x^2}$

$\int \frac{dt}{t^2+1}=\int \frac{dt}{t^2+1^2}$

Here $x=t,a=1$

$=\frac{1}{1}ta{n}^{-1}\frac{t}{1}+C$

$=ta{n}^{-1}t+C$ $[\because t\because =x^3]$

$=ta{n}^{-1}{x}^3+C$