Evaluate the following integrals: ^ 1_ -1 5x^4 x^5+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits1_{-1}5\text{x}^4\sqrt{\text{x}^5+1}\text{ dx}$

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Answer

Let $\text{I}=\int^\limits1_{-1}5\text{x}^4\sqrt{\text{x}^5+1}\text{ dx}$ Then,
Let $\text{x}^5+1=\text{t}$ Then, $5\text{x}^4\text{ dx}=\text{dt}$
When $\text{x}=-1,\text{t}=0$ and $\text{x}=1,\text{t}=2$
$\therefore\ \text{I}=\int\limits^2_0\sqrt{\text{t}}\text{ dt}$
$\Rightarrow\text{I}=\Big[\frac{2}{3}\text{t}^{\frac{3}{2}}\Big]^6_0$
$\Rightarrow\text{I}=\frac{2}{3}\sqrt{8}$
$\Rightarrow\text{I}=\frac{4\sqrt{2}}{3}$

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