Question

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

Answer 1

`\bf \displaystyle \int x^2(x^3+9)^{(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)^{(1)/(2)}\cdot \cfrac{du}{3\underline{x^2}}\implies \int \cfrac{u^{(1)/(2)}}{3}\cdot du\implies \cfrac{1}{3}\int u^{(1)/(2)}\cdot du \\\\\\ \cfrac{1}{3}\cdot \cfrac{u^{(3)/(2)}}{(3)/(2)}\implies \cfrac{1}{3}\cdot \cfrac{2√(u^3)}{3}\implies \cfrac{2√(u^3)}{9}\implies \cfrac{2√((x^3+9)^3)}{9}+C`