Mathematics

Evaluate: $$\displaystyle \int_{0}^{1}\frac{x^{3}}{1+x^{8}} dx$$ 


ANSWER

$$\displaystyle \frac{\pi}{16}$$


SOLUTION
Let, $$z=x^4\implies dz=4x^3dx\implies x^3dx=\dfrac{dz}{4}$$
 
When $$ x=0,z=0$$ and when $$x=1,z=1$$

Hence,integration becomes:-

$$\displaystyle\dfrac{1}{4}\int_{0}^{1} \dfrac{dz}{1+z^2}$$

$$ =\dfrac{1}{4}\displaystyle\left[\tan^{-1}{z}\right]_{0}^{1}$$

$$=\dfrac{1}{4}\displaystyle\left[\dfrac{\pi}{4}-0\right]$$

$$=\dfrac{\pi}{16}$$

Hence, answer is option-(C).
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