Math Help - help with integration!

1. help with integration!

Really need help! How do I integrate 4/(sqrt(x)sqrt(1-x)) and e^(-x)cosx? I tried using algebraic manipulation for the first one and integration by parts for the second, got a mess of workings, really will appreciate your help!

2. Re: help with integration!

Hey Tutu.

Try using the fact that SQRT(x)*SQRT(1-x) = SQRT(x - x^2) and then complete the square.

Hint: Completing the square for ax^2 + bx is adding the term (0.5*b/SQRT(a))^2 and also -(0.5*b/SQRT(a))^2 to get (SQRT(a)*x + 0.5*b/SQRT(a))^2 - (0.5*b/SQRT(a))^2.

3. Re: help with integration!

Thanks! Couof you help ke with also the second one, thanks so much!

4. Re: help with integration!

For the second on try integration by parts by integrating the the trigonometric term twice (in other words do by parts twice) and then collect terms.

5. Re: help with integration!

Solution for the second integral is given in the attached sheet.
Be careful in selecting the first and second function while integrating by parts.

6. Re: help with integration!

Hi thanks a lot for both your help! I am confused though, according to the LIATE rule,
u should be cos(x) since trigo functions come before exponential, isn't it?

7. Re: help with integration!

In this case it won't make any difference, as exponential functions only change by a constant multiple when differentiated or integrated, and sine and cosine functions repeat each other by constant multiples after being differentiated or integrated twice.