Evaluate _ -1 ^ 1 5 x^ 4 x^ 5 +1 d x - Vidyadip

Question

Evaluate $\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d\ x$

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Answer

Put $t = x^5 + 1,$ then $dt = 5x^4 d\ x.$
Therefore, $\int 5 x^{4} \sqrt{x^{5}+1} d\ x$
$= \int \sqrt{t} d\ t$
$=\frac{2}{3} t^{\frac{3}{2}}$
$=\frac{2}{3}\left(x^{5}+1\right)^{\frac{3}{2}}$
Hence, $\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x$
$= \frac{2}{3}\left[\left(x^{5}+1\right)^{\frac{3}{2}}\right]_{-1}^{1}$
$= \frac{2}{3}\left[\left(1^{5}+1\right)^{\frac{3}{2}}-\left((-1)^{5}+1\right)^{\frac{3}{2}}\right]$
$= \frac{2}{3}\left[2^{\frac{3}{2}}-0^{\frac{3}{2}}\right]$
$=\frac{2}{3}(2 \sqrt{2})$
$=\frac{4 \sqrt{2}}{3}$

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