Definite integral problem. I don't see a clear u-sub

Question

I have this integral:

$$ \int_1^2 x \cdot \sqrt{x-1} \, dx$$

I don't see it the clear u sub. If I take $ = x - 1$ then the derivative is 1 and that isn't substituable in the integrand.

Answer 1

Why would $u=x-1$ not be suitable? Before you make any assumptions, try it first

$$ \int x\sqrt{x-1}\ dx = \int (u+1)\sqrt{u}\ du = \int u^{3/2} + u^{1/2}\ du = \frac{2}{5}u^{5/2} + \frac{2}{3}u^{3/2} + C $$

You can see that the substitution works.

Answer 2

Hint:

Let $x-1=u^2$, then $x=u^2+1$