Question

Solve $\int x^5\cdot \sqrt{a^3+x^3}dx$

Answer 1

$I=\int x^5\cdot \sqrt{a^3+x^3}dx$

$=\int x^3\cdot x^2\sqrt{a^3+x^3}dx$

Put $a^2+x^3=t^2$

$\therefore t=(a^4+x^3{)}^{\frac{1}{2}}$

$\Rightarrow 3x^{2}dx=2tdt$

$\Rightarrow x^{2}dx=\frac{2t}{3}dt$

$=\int (t^2-a^3)\cdot \frac{2t}{3}\cdot tdt$

$=\int (t^2-a^3)\frac{2t^2}{3}dt$

$=\frac{2}{3}\int (t^4-t^2{a}^3)dt$

$=\frac{2}{3}[\frac{t^5}{5}-\frac{t^3}{3}{a}^3]+c$

$=\frac{2}{3}[\frac{(a^3+x^3{)}^{\frac{5}{2}}}{5}-\frac{a^3}{3}(a^3+x^3{)}^{\frac{3}{2}}]+c$.