Mathematics

# Evaluate $\displaystyle \int \dfrac {x^{3}}{\sqrt {1+2x^4}}dx$

##### SOLUTION
We have,
$I=\displaystyle \int \dfrac {x^{3}}{\sqrt {1+2x^4}}dx$

Let
$t=1+2x^4$
$\dfrac{dt}{dx}=0+8x^3$
$\dfrac{dt}{8}=x^3\ dx$

Therefore,
$I=\dfrac{1}{8}\displaystyle \int \dfrac {1}{\sqrt {t}}dt$

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

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

On putting the value of $t$, we get
$I=\dfrac{1}{4}\left[\sqrt {1+2x^4}\right]+C$

Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
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