Question

The value of $\int \frac{x^{2}dx}{\sqrt{x^6+a^6}}$ 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)

$x^3=t3x^{2}dx=dt$
$I=\frac{1}{3}\int \frac{1}{\sqrt{t^2+{\left(a^3\right)}^2}}dt$
$=\frac{1}{3}\log \left[t+\sqrt{t^2+a^6}\right]$
$=\frac{1}{3}\log \left[x^3+\sqrt{x^6+a^6}\right]+c$