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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