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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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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