Integrate the following functions w.r.t. x: x^5 a^2+x^2

Question

Integrate the following functions w.r.t. x:
$x^5 \sqrt{a^2+x^2}$

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Answer

Let $I=\int x^5 \sqrt{a^2+x^2} d x$Put, $a^2+x^2=t$
$\therefore 2 x d x=d t \quad \therefore x d x=\frac{1}{2} d t$
Also, $x^2=t-a^2$
$I =\int x^2 \cdot x^2 \sqrt{a^2+x^2} x d x$
$=\frac{1}{2} \int\left(t-a^2\right)^2 \sqrt{t} d t$
$=\frac{1}{2} \int\left(t^2-2 a^2 t+a^4\right) \sqrt{t} d t$
$=\frac{1}{2} \int\left(t^{\frac{5}{2}}-2 a^2 t^{\frac{3}{2}}+a^4 t^{\frac{1}{2}}\right) d t$
$=\frac{1}{2} \int t^{\frac{5}{2}} d t-a^2 \int t^{\frac{3}{2}} d t+\frac{a^4}{2} \int t^{\frac{1}{2}} d t$
$=\frac{1}{2} \cdot \frac{t^{\frac{7}{2}}}{\left(\frac{7}{2}\right)}-a^2 \cdot \frac{t^{\frac{5}{2}}}{\left(\frac{5}{2}\right)}+\frac{a^4}{2} \cdot \frac{t^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}+c$
$=\frac{1}{7}\left(a^2+x^2\right)^{\frac{7}{2}}-\frac{2 a^2}{5}\left(a^2+x^2\right)^{\frac{5}{2}}+\frac{a^4}{3}\left(a^2+x^2\right)^{\frac{3}{2}}+c .$

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