Section 5.2. Will somebody verify this solution for me?
Evalute $\int 5x^4 \sqrt{x^5+5}dx$ using u substitution
Let $u=x^5+5$. Then $\frac{d}{dx}u=\frac{du}{dx}=\frac{d}{dx}(x^5+5)=5x^4$.
Making these substitutions into the original integral, simplifying, and then evaluating the integral, we get:
$\int \frac{du}{dx} \sqrt{u} dx = \int \sqrt{u} \frac{du}{dx} dx = \int \sqrt{u} \frac{dx}{dx} du = \int \sqrt{u} du$
$= \int u^{\frac{1}{2}}du=\frac{u^{\frac{3}{2}}}{\frac{3}{2}}+C=\frac{2}{3}u^{\frac{3}{2}}+C = \frac{2}{3}(x^5+5)^{\frac{3}{2}}+C $
That answer is correct and the work is correct (but a bit excessive).