Integrating x/(1-4x-2x2)1/2: Different Approaches

[gaobo9109]

Homework Statement

How do you integrate x/(1-4x-2x²)^1/2

Homework Equations

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 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.