Integral problem

Question

solve for the integral.......cos(x)sqrt(1-cos(x)) in respects to x, lower bound is o and upper bound is pi/2

Answer 1

∫cos(x)√(1-cos(x) dx= ∫cos(x)√(1-cos²x)/√(1+cos(x) dx = ∫cos(x) sin(x) /√(1+cos(x) dx

let: u= 1+cos(x) then du=-sin(x) dx

Thus, the integral becomes: ∫(u-1)/u^1/2 (-du) = -∫ (u^1/2 - u^-1/2) du = -(2/3)u^3/2 + 2u^1/2 + C, which after substitution becomes: -(2/3)(1+cos(x))^3/2 + 2(1+cos(x))^1/2 + C

 

Since it is a definite integral, plug the limits, the answer should be (-2/3+2)-(-2/3)(2√2)-(2√2) = 4/3 -2√2/3