Browse Questions

Integrate the rational functions$\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$

$\begin{array}{1 1}x-\frac{2}{3}\tan^{-1}\big(\frac{x}{\sqrt 3}\big)+3\tan^{-1}\big(\frac{x}{2}\big)+c. \\x+\frac{2}{3}\tan^{-1}\big(\frac{x}{\sqrt 3}\big)-3\tan^{-1}\big(\frac{x}{2}\big)+c. \\ x+\frac{2}{3}\tan^{-1}\big(\frac{x}{\sqrt 3}\big)+3\tan^{-1}\big(\frac{x}{2}\big)+c. \\ \frac{2}{3}\tan^{-1}\big(\frac{x}{\sqrt 3}\big)+3\tan^{-1}\big(\frac{x}{2}\big)+c.\end{array}$

Toolbox:
• If f(x)=t,then on differentiating f'(x)dx=dt.
• Thus $\int f(x)dx=\int tdt.$
• (ii)If the rational function is improper in nature.we can divide and separate the terms to make it a proper rational function.
• (iii)$\int\frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\big(\frac{x}{a}\big)+c.$
• (iv)$\int\frac{dx}{x+a}=log|x+a|+c.$
Given $I=\int\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}.$