Solving Integrals: x/√(1-x)

[noblerare]

[SOLVED] Several Integrals

Homework Statement

$$\int$$$$\frac{xdx}{\sqrt{1-x}}$$

Homework Equations

n/a

The Attempt at a Solution

I tried to multiple top and bottom by the conjugate:

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

After trig substitution, I end up with:

-$$\int$$cos$$\theta$$$$\sqrt{1+cos\theta}$$d$$\theta$$

I've also tried u-substitution but nothing seems to work.

Can anyone help me out on this? And show steps, please? Thanks so much!

 

[lippyka]

Solution: We can use the substitution u=1-x, which yields\int \frac{(1-u)du}{\sqrt{u}} = \int \frac{u-1}{\sqrt{u}}duNow, we can split up the integral into two integrals\int u^{1/2}du - \int \frac{1}{\sqrt{u}}duThe first integral is easy to solve. We have\frac{2}{3}u^{3/2} + C_1For the second integral, we use the substitution v=1/u, which yields-2\sqrt{v} + C_2Substituting back for v and u respectively yields-2\sqrt{1/u} + C_2 = -2\sqrt{1/(1-x)} + C_2Our final solution is thus\frac{2}{3}u^{3/2} - 2\sqrt{1/(1-x)} + C