Question

Evaluate:
${\int }_{-1}^{1}5x^4\sqrt{x^5+1}dx$

Answer 1

${\int }_{-1}^{1}5x^4\sqrt{x^5+1}dx$

Let,

$F\left(x\right)=\int 5x^4\sqrt{x^5+1}dx$

Putting $t=x^5+1$

Differentiating \9w.r.t.x\)

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

On substituting

$\int 5x^4\sqrt{x^5+1}dx$

$=\int \sqrt{t}dx$

$=\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}$

$=\frac{2}{3}{t}^{\frac{3}{2}}$

$F\left(x\right)=\frac{2}{3}(x^5+1{)}^{\frac{3}{2}}[\because t=x^5+1]$

Now,

${\int }_{-1}^{1}5x^4\sqrt{x^5+1}dx=F\left(1\right)-F(-1)$

$=\frac{2}{3}(1^5+1{)}^{\frac{3}{2}}-\frac{2}{3}((-1{)}^5+1{)}^{\frac{3}{2}}$

$=\frac{2}{3}(1+1{)}^{\frac{3}{2}}-\frac{2}{3}(-1+1{)}^{\frac{3}{2}}$

$=\frac{2}{3}(2{)}^{\frac{3}{2}}-0$

$=\frac{2}{3}\times 2\sqrt{2}$

$=\frac{4\sqrt{2}}{3}$