Mathematics

Evaluate the following definite integral:

$$\displaystyle\int_{-1}^{1}5x^{4}\sqrt{x^{5}+1}\ dx$$


SOLUTION

Let $$I=\displaystyle\int_{-1}^{1}5x^{4}\sqrt{x^{5}+1}\ dx$$ 

let $$x^{5}+1=t$$. 

Then, $$d(x^{5}+1)=dt$$ 

$$\Rightarrow 5x^{4}dx=dt$$

Also, 

$$x=-1\Rightarrow t=0$$

$$x=1\Rightarrow t=2$$

$$\therefore I=\displaystyle\int_{0}^{2}\sqrt{t}dt$$ 

$$I=\dfrac{2}{3}\left[t^{3/2}\right]_{0}^{2}$$ 

$$I=\dfrac{2}{3}\times 2^{3/2}$$ 

$$I=\dfrac{4\sqrt{2}}{3}$$

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