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!
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.
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.