Integration With Completing the Square

[Bashyboy]

The problem is: $\int\frac{1}{\sqrt{1-4x-x^2}}dx$

I took the expression under the radical and I completed the square, yielding: $\int\frac{1}{\sqrt{5-(x+2)^2}}dx$

Then I figured that I could apply the arcsin formula, where $a^2=5$ and$u^2=(x+2)^2$

But by solving for "a" and "u," I would be left with two roots, wouldn't I?

 

[voko]

What exactly would you solve for a and u?

 

[christoff]

Forget about the "formula". If you're unsure, do some integration by substitution and try to get to something familiar.

$$\int \frac{1}{\sqrt{5-(x+2)^2}}dx=\int \frac{1}{\sqrt{5}}\frac{1}{\sqrt{1-\frac{(x+2)^2}{5}}}dx=\frac{1}{\sqrt{5}} \int \frac{1}{\sqrt{1-\left(\frac{x+2}{\sqrt{5}}\right)^2}}dx$$
What should you do next? The integrand is essentially $\frac{1}{\sqrt{1-x^2}}$, you just need the right substitution.

 

[Bashyboy]

Thank you, Christoff, that was rather clever.