Question

Evaluate
$\int \frac{x^3}{\sqrt{1+2x^4}}dx$

Answer 1

We have,

$I=\int \frac{x^3}{\sqrt{1+2x^4}}dx$

Let

$t=1+2x^4$

$\frac{dt}{dx}=0+8x^3$

$\frac{dt}{8}=x^{3}dx$

Therefore,

$I=\frac{1}{8}\int \frac{1}{\sqrt{t}}dt$

$I=\frac{1}{8}\left[2\sqrt{t}\right]+C$

$I=\frac{1}{4}\left[\sqrt{t}\right]+C$

On putting the value of $t$, we get

$I=\frac{1}{4}\left[\sqrt{1+2x^4}\right]+C$

Hence, this is the answer.