Question

$\displaystyle \int \frac{x^{n-1}}{x^{2n}+a^{2}} dx $dX is equal to

• A. $\frac{1}{na} tan^{-1} \left(\frac{x^{n}}{a}\right) +C$
• B. $\frac{n}{a} tan^{-1} \left(\frac{x^{n}}{a}\right) +C$
• C. $\frac{n}{a} sin^{-1} \left(\frac{x^{n}}{a}\right) +C$
• D. $\frac{n}{a} cos^{-1} \left(\frac{x^{n}}{a}\right) +C$
• E. $\frac{1}{na} cot^{-1} \left(\frac{x^{n}}{a}\right) +C$

Answer 1

Answer: (A)

Let $I=\int \frac{x^{n-1}}{x^{2 n}+a^{2}} d x$
$\operatorname{Put} x^{n}=t$
$\Rightarrow n x^{n-1} d x=d t$
$\therefore I=\frac{1}{n} \int \frac{d t}{t^{2}+a^{2}}=\frac{1}{n} \cdot \frac{1}{a} \tan ^{-1} \frac{t}{a}+C$
Put $t=x^{n}$
$\Rightarrow I=\frac{1}{n a} \tan ^{-1}\left(\frac{X^{n}}{a}\right)+C$