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  • Evaluate the integral of <img alt="\int\frac{x^{4}dx}{\sqrt{x^{10}-1}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cint%5Cfrac%7Bx%5E%7B4%7Ddx%7D%7B%5Csqrt%7Bx%5E%7B10%7D-1%7D%7D"

Evaluate the integral of \int\frac{x^{4}dx}{\sqrt{x^{10}-1}}

Option: 1

-\frac{1}{5}log\left | x^{5}-\sqrt{x^{10}-1} \right |+C


Option: 2

\frac{1}{5}log\left | x^{5}-\sqrt{x^{10}-1} \right |+C


Option: 3

-\frac{1}{5}log\left | x^{5}+\sqrt{x^{10}-1} \right |+C


Option: 4

\frac{1}{5}log\left | x^{5}+\sqrt{x^{10}-1} \right |+C


Answers (1)

best_answer

Given that,
\int\frac{x^{4}dx}{\sqrt{x^{10}-1}}
 

Substitute,
x^{5}=t\Rightarrow 5x^{4}dx=dt
x^{4}dx=\frac{dt}{5}
Thus,
\int\frac{x^{4}dx}{\sqrt{x^{10}-1}}=\frac{1}{5}\int \frac{dt}{\sqrt{t^{2}-1^{2}}}


Using the integral formula,


\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=log\left | x+\sqrt{x^{2}-a^{2}} \right |+C


Therefore,


\int\frac{x^{4}dx}{\sqrt{x^{10}-1}}=\frac{1}{5}log\left | t+\sqrt{t^{2}-1} \right |+C
\int\frac{x^{4}dx}{\sqrt{x^{10}-1}}=\frac{1}{5}log\left | x^{5}+\sqrt{x^{10}-1} \right |+C

Posted by

manish

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