Question

Integrate the following:

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

ii) $\int \frac{1}{{\cos }^{4}x+{\sin }^{4}x}dx$

Answer 1

i)

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

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

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

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

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

ii)

$\int \frac{1}{{\cos }^{4}x+{\sin }^{4}x}dx$

$\int {\sec }^{2}x\frac{{\tan }^{2}x+1}{{\tan }^{4}x+1}dx$

$\frac{{\tan }^{-1}(\sqrt{2}\tan x+1)}{\sqrt{2}}+\frac{{\tan }^{-1}(\sqrt{2}\tan x-1)}{\sqrt{2}}$

$\int \frac{1}{{\cos }^{4}x+{\sin }^{4}x}dx=\frac{{\tan }^{-1}(\sqrt{2}\tan x+1)+{\tan }^{-1}(\sqrt{2}\tan x-1)}{\sqrt{2}}C$