Integration question

Homework Statement

How do you integrate x/(1-4x-2x2)1/2

The Attempt at a Solution

I tried to solve it using by-part method, but it turned out to be very complicated, and my answer doesn't match the answer given. Is there any other way to approach this question?

LCKurtz
Homework Helper
Gold Member

Homework Statement

How do you integrate x/(1-4x-2x2)1/2

I tried to solve it using by-part method, but it turned out to be very complicated, and my answer doesn't match the answer given. Is there any other way to approach this question?

If you let u = 1 - 4x -2x2 then du would be -4 - 4x. So what I would try first is to build that into the numerator because it has an x already:

$$\int \frac x {\sqrt{1-4x-2x^2}}\,dx =-\frac 1 4 \int \frac {-4x}{\sqrt{1-4x-2x^2}}\, dx =-\frac 1 4 \int \frac {-4x - 4 + 4}{\sqrt{1-4x-2x^2}}\, dx =-\frac 1 4 \int \frac {-4x-4}{\sqrt{1-4x-2x^2}} +\frac 4 {\sqrt{1-4x-2x^2}}\, dx$$

Then I would do the u-substitution on the first and for the second I would try completing the square under the integral looking for an arcsine form.