Question

The value of $\int \frac{x^{2} d x}{\sqrt{x^{6}+a^{6}}}$ is equal to

• A. $\log \left| x ^{3}+\sqrt{ x ^{6}+ a ^{6}}\right|+ c$
• B. $\log \left|x^{3}-\sqrt{x^{6}+a^{6}}\right|+c$
• C. $\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+c$
• D. $\frac{1}{3} \log \left|x^{3}-\sqrt{x^{6}+a^{6}}\right|+c$

Answer 1

Answer: (C)

$x ^{3}= t \,\,\,3 x ^{2} d x = dt$
$I =\frac{1}{3} \int \frac{1}{\sqrt{ t ^{2}+\left( a ^{3}\right)^{2}}} dt$
$=\frac{1}{3} \log \left[ t +\sqrt{ t ^{2}+ a ^{6}}\right]$
$=\frac{1}{3} \log \left[ x ^{3}+\sqrt{ x ^{6}+ a ^{6}}\right]+ c$