Question

If $I=\int \frac{x^2+a^2}{x^4-a^2{x}^2+a^4}dx$

• A. $\frac{1}{a}{\tan }^{-1}\left(\frac{ax}{x^2-a^2}\right)+C$
• B. $\frac{1}{a}{\tan }^{-1}\left(\frac{x^2-a^2}{ax}\right)+C$
• C. $\log |x+\sqrt{x^2-a^2}|+x+C$
• D. none of these

Answer 1

The correct option is B $\frac{1}{a}{\tan }^{-1}\left(\frac{x^2-a^2}{ax}\right)+C$

$I=\int \frac{x^2+a^2}{x^4-a^2{x}^2+a^4}dx=\int \frac{1+\frac{a^2}{x^2}}{x^2+\frac{a^4}{x^2}-a^2}dx$
Put $1-\frac{a^2}{x^2}=t$
$I=\int \frac{dt}{t^2+a^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{t}{a}\right)+c=\frac{1}{a}{\tan }^{-1}\left(\frac{x^2-a^2}{ax}\right)+c$