Use Substitution or partial integration to solve $\int x\sqrt{1-x^2}\,dx$

Question

I'm struggling with the following homework:

Use Substitution or partial integration to solve

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

Ok, so since we have a multiplication, partial integration seems like the right choice:

$$\int f'(x)g(x)\,dx = f(x)g(x)-\int f(x)g'(x)\,dx$$

$$f'(x) = x, g(x) = \sqrt{1-x^2} \Rightarrow f(x) = \frac{1}{2}x^2, g'(x) = -\frac{x}{\sqrt{1-x^2}}$$

So I end up with

$$\frac{1}{2}x^2 \cdot \sqrt{1-x^2} - \int \frac{-x^3}{2\sqrt{1-x^2}}\,dx$$

But this is in no way easier to integrate than the expression I started with. So where am I going wrong?

Answer 1

$$ \int\sqrt{1-x^2}\Big( x\,dx \Big) $$

Writing these huge parentheses around $x\,dx$ should be taken as a hint. It tells you what substitution to use. If you don't perceive that as a hint, then you need to work on your understanding of how integration by substitution works.

Answer 2

Hint:
Start from this: $2x\,dx=d(x^2)$ and note that $d(1-x^2)=-2x\,dx.$