Question

Evaluate the integral by making the given substitution. (Use C for the constant of integration.) *? Vx2 + 7 dx, u = x3 + 7

Answer 1

we are given

$\int x^2\sqrt{x^3+7}dx$

we can use u-subs

$u=x^3+7$

$du=3x^2dx$

$=\int \frac{\sqrt{u}}{3}du$

$=\frac{1}{3}\cdot \int \:u^{\frac{1}{2}}du$

we can use power rule

$=\frac{1}{3}\cdot \frac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}$

now, we can plug back u

and we get

$=\frac{2}{9}\left(x^3+7\right)^{\frac{3}{2}}+C$.............Answer