Question

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

Answer 1

$\int \frac{x^2}{\left(x^2+1\right)\left(x^2+4\right)}dx$

$Let{x}^2=t$

$=\int \frac{t}{\left(t+1\right)\left(t+4\right)}dx$

$=\int \frac{\frac{4}{3}}{t+4}dx-\int \frac{\frac{1}{3}}{t+1}dx$

$=\frac{4}{3}\int \frac{dx}{d+4}-\frac{1}{3}\int \frac{dx}{t+1}$

$=\frac{4}{3}\int \frac{dx}{x^2+{\left(2\right)}^2}-\frac{1}{3}\int \frac{dx}{x^2+1}$

$=\frac{4}{3}\times \frac{1}{2}{\tan }^{-1}\frac{x}{2}-\frac{1}{3}{\tan }^{-1}x+c$

$=\frac{2}{3}{\tan }^{-1}\frac{x}{2}-\frac{1}{3}{\tan }^{-1}x+c$