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) \[\frac{n}{a}{{\cos }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\]
Correct Answer: A
Solution :
[a] Let \[I=\int{\frac{{{x}^{n-1}}dx}{{{x}^{2n}}+{{a}^{2}}}}\] Let \[{{x}^{n}}=t\Rightarrow n\,\,.\,\,{{x}^{n-1}}dx=dt\] \[\therefore I=\int{\frac{1}{n}.\frac{dt}{{{t}^{2}}+{{a}^{2}}}=\frac{1}{n}\,\,.\,\,\frac{1}{a}{{\tan }^{-1}}\left( \frac{t}{a} \right)+C}\] \[=\frac{1}{na}{{\tan }^{-1}}\left[ \frac{{{x}^{n}}}{a} \right]+C\]
You need to login to perform this action.
You will be redirected in
3 sec
