Mathematics

# Evaluate $\displaystyle \int\cfrac{x^{2}}{\sqrt{x^{6}+a^{6}}} \ dx$

##### SOLUTION
Let $I=\displaystyle \int \cfrac{x^{2}}{\sqrt{x^{6}+a^{6}}} dx$
Put $x^3=t$
$3x^2 dx=dt$
$x^2 dt=dt/3$
$I=\displaystyle \int \dfrac{x^2}{\sqrt{(x^3)^2+a^6}} dx$
$=\dfrac{1}{3} \displaystyle \int \dfrac{dt}{\sqrt{t^2+(a^3)^2}}dx$
$=\dfrac{1}{3} \times \log |t+ \sqrt{t^2+(a^3)^2}|+C$
$=\displaystyle \dfrac{1}{3} \log |x^3+ \sqrt{x^6+(a^6}|+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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