1. ## Proving with Integration?

I was studying and I met the following problem:

Prove that $\int\sqrt{1-(a^2)(x^2)}$(dx)= $\sqrt{1-(a^2)}$ $+\frac{\arcsin(a)}{a}$ for 0 < a < 1

The integration has limits from -1 to 1. (Don't know how to show them with Latex ) And its in terms of x. (i.e: (dx))

That "a" is causing me a lot of problems. If anyone can work out this proof for me so that I could retry it, I would deeply appreciate it.

2. Originally Posted by Tick
I was studying and I met the following problem:

Prove that $\int\sqrt{1-(a^2)(x^2)}$(dx)= $\sqrt{1-(a^2)}$ $+\frac{\arcsin(a)}{a}$ for 0 < a < 1

The integration has limits from -1 to 1. (Don't know how to show them with Latex ) And its in terms of x. (i.e: (dx))

That "a" is causing me a lot of problems. If anyone can work out this proof for me so that I could retry it, I would deeply appreciate it.

$\int_{-1}^{1} \sqrt{1-a^2x^2} dx = \int_{-1}^{1} \sqrt{a^2\bigg(\frac{1}{a^2}-x^2\bigg)} dx$

$\int_{-1}^{1} a\sqrt{\bigg(\frac{1}{a}\bigg)^2 -x^2} dx = a\int_{-1}^{1} \sqrt{\bigg(\frac{1}{a}\bigg)^2 -x^2} dx$