Question

Evaluate the Integral \(\int {\frac{{{x^4}dx}}{{\sqrt {{x^{10}} - 2} }}} \).

Short answer

Substitute \({x^5} = t\) and then integrate it. After integration, again put \(t = {x^5}\)

The value of the integral is \(\frac{1}{5}ln\left| {{x^5} + \sqrt {{x^{10}} - 2} } \right| + c\).

Step-by-step solution

Substitute \({x^5} = t\).

Let \({x^5} = t\).

\(\begin{aligned}{l}5{x^4}dx = dt\\{x^4}dx = \frac{{dt}}{5}\end{aligned}\)

\(\therefore \int {\frac{{{x^4}dx}}{{\sqrt {{x^{10}} - 2} }} = \frac{1}{5}\int {\frac{{dt}}{{\sqrt {{t^2} - 2} }}} } \)

Evaluate the integral.

Using the formula, \(\int {\frac{1}{{\sqrt {{x^2} - {a^2}} }}dx = \ln \left| {x + \sqrt {{x^2} - {a^2}} } \right| + c} \).

\(\frac{1}{5}\int {\frac{1}{{\sqrt {{t^2} - 2} }}dx = \frac{1}{5}ln\left| {t + \sqrt {{t^2} - 2} } \right| + c} \)

Hence, \(\int {\frac{{{x^4}}}{{\sqrt {{x^{10}} - 2} }}dx = \frac{1}{5}ln\left| {{x^5} + \sqrt {{x^{10}} - 2} } \right| + c} \).