If f is continuous, prove the equation below and evaluate.

Question

Part 1: if f is continuous, prove that

(0,π/2) ∫f(cosx)dx=(0,π/2) ∫f(sinx)dx

Part 2: Use the exercise above to evaluate

(0,π/2) ∫cos²(x)dx=(0,π/2) ∫sin²(x)dx

Answer 1

This Desmos graph sort of confirms the proof in the 1st part of the video.

desmos.com/calculator/9csomvfets

This Desmos graph confirms the 2nd part:

desmos.com/calculator/6lwyf4idko

Answer 2

On the left make the u substitution u=cos x and on the right make the u substitution u=sin x.

The integrand on the left becomes -f(u) du/sqrt(1-u²) and on the right becomes f(u) dy/sqrt(1-u^2)

The limits on the left and 1 to 0 and the right are 0 to 1.

And these are equal.

This tells you that the equation in the 2nd part is true.

To evaluate use the identity cos² x= .5(1+cos 2x).