Question

Evaluate the integral x^2(x^3+9)^1/2

Answer 1

$\bf \displaystyle \int x^2(x^3+9)^{\frac{1}{2}}\cdot dx\\\\ -------------------------------\\\\ u=x^3+9\implies \cfrac{du}{dx}=3x^2\implies \cfrac{du}{3x^2}=dx\\\\ -------------------------------$

$\bf \displaystyle \int \underline{x^2}(u)^{\frac{1}{2}}\cdot \cfrac{du}{3\underline{x^2}}\implies \int \cfrac{u^{\frac{1}{2}}}{3}\cdot du\implies \cfrac{1}{3}\int u^{\frac{1}{2}}\cdot du \\\\\\ \cfrac{1}{3}\cdot \cfrac{u^{\frac{3}{2}}}{\frac{3}{2}}\implies \cfrac{1}{3}\cdot \cfrac{2\sqrt{u^3}}{3}\implies \cfrac{2\sqrt{u^3}}{9}\implies \cfrac{2\sqrt{(x^3+9)^3}}{9}+C$