How to use integration by parts to solve an integral? I'm having trouble with solving this integral by parts. Here is what I have so far. Can anyone please help me out?
Solve the integral $\int x^3\sqrt{x^2-1}dx$ by parts, choosing u = $x^2$ and
$dv = x\sqrt{x^2-1} dx$
$du_1 = 2xdx$
$v_1$ = $\int x\sqrt{x^2-1}dx$ $ t = x^2-1$ $dt = 2x dx$
$v_1$ = $\frac{1}{2}\int \sqrt{t}dt$
$v_1$ = $\frac{1}{2}(\frac{2}{3}t^\frac{3}{2}) + c$
$v_1$ = $\frac{1}{3} (x^2-1)^\frac{3}{2} + c$
$uv - \int vdu$
= $\frac{1}{3} (x^2-1)^\frac{3}{2}x^2 - \int 2x\frac{1}{3} (x^2-1)^\frac{3}{2}dx$
$u_2 = 2x$ $dv_2 = \frac{1}{3} (x^2-1)^\frac{3}{2}dx$
$du_2 = 2dx$ 

You used integration by parts once. And you used substitution to find $v_1$. Use substitution again:
$$I=\int 2x\cdot \frac13(x^2-1)^{3/2}dx;$$
$$t=x^2-1; \ \ dt=2xdx;$$
$$I=\frac13 \int t^{3/2}dt=\frac13\cdot \frac25 \cdot t^{5/2}+C=\frac{2}{15}(x^2-1)^{5/2}+C.$$
