Question

$\int \frac{x^{n-1}}{x^{2n}+a^2}dx$ 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: (B)

$\frac{n}{a}{\tan }^{-1}\left(\frac{x^n}{a}\right)+C$

Let $I=\int \frac{x^{n-1}}{x^{2n}+a^2}dx$
Put $x^n=t$
$\Rightarrow n{x}^{n-1}dx=dt$
Therefore, $I=\frac{1}{n}\int \frac{dt}{t^2+a^2}$

$=\frac{1}{n}\cdot \frac{1}{a}{\tan }^{-1}\frac{t}{a}+C$
Put $t=x^n$
$\Rightarrow I=\frac{1}{na}{\tan }^{-1}\left(\frac{x^n}{a}\right)+C$