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

Question

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

✓

Answer

Let
$
I=\int_{-1}^1 5 x^4 \sqrt{x^5+1} d x
$
Putting $t=x^5+1 \Rightarrow d t=5 x^4 d x$
So,
$
\begin{aligned}
\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}}
\end{aligned}
$
So, $\int_{-1}^1 5 x^4 \sqrt{x^5+1} d x$
$
\begin{array}{l}
=\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}-0) \\
=\frac{4 \sqrt{2}}{3} \text { }
\end{array}
$

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