Question

If $\int \sqrt{\frac{x^4}{a^6+x^6}}dx=g\left(x\right)+C,$ then $g\left(x\right)$ equals to (where $C$ is constant of integration)

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

Answer 1

The correct option is C $\frac{1}{3}\log |x^3+\sqrt{a^6+x^6}|$

We have, $I=\int \sqrt{\frac{x^4}{a^6+x^6}}dx\Rightarrow I=\int \frac{x^2}{\sqrt{(a^3{)}^2+(x^3{)}^2}}dx$
Let $x^3=t\Rightarrow 3x^{2}dx=dt$
$I=\frac{1}{3}\int \frac{1}{\sqrt{(a^3{)}^2+t^2}}dt=\frac{1}{3}\log |t+\sqrt{t^2+a^6}|+C[\because \int \frac{1}{\sqrt{a^2+x^2}}dx=\log |x+\sqrt{a^2+x^2}|+c]$
$\Rightarrow I=\frac{1}{3}\log |x^3+\sqrt{x^6+a^6}|+C$
On comparing with given relation, we get $g\left(x\right)=\frac{1}{3}\log |x^3+\sqrt{a^6+x^6}|$