Question

Solve $\int \frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}dx$

Answer 1

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

This can be written as

$\Rightarrow \int (\frac{-4x^2-10}{(x^2+3)(x^2+4)}+1)dx$

$\Rightarrow \int \frac{-4x^2-10}{(x^2+3)(x^2+4)}dx+\int 1dx$

$\Rightarrow -2\int \frac{2x^2+5}{(x^2+3)(x^2+4)}dx+\int 1dx$

$\Rightarrow -2\int \frac{3}{x^2+4}dx+2\int \frac{1}{x^2+3}dx+x$

$\Rightarrow -6\int \frac{1}{x^2+4}dx+2\int \frac{1}{x^2+3}dx+x$

$\Rightarrow -\frac{6}{2}{\tan }^{-1}\left(\frac{x}{2}\right)+\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{x}{\sqrt{3}}\right)+x$

$\Rightarrow x-3{\tan }^{-1}\left(\frac{x}{2}\right)+\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{x}{\sqrt{3}}\right)$