Using Wallis’ Formula

Question

a) Evaluate the integral of 4sin⁶xcos⁴xdx using lower limit = 0 and upper limit =π/2.

b) Find the value of the integral of cos⁸3xdx using the limits 0 to π/6.

Answer 1

A. ∫ sin⁶xcos⁴xdx =∫sin⁶x (1- sin²x )²dx = ∫ sin⁶x ( 1 - 2sin²x + sin⁴x ) dx = ∫ sin⁶x dx -2 ∫sin⁸x dx +∫sin¹⁰x dx

and then use the reduction formulas

B. You can also use the reduction formula

∫sin^mx cosⁿx dx = [- (sin^m-1x cosⁿ⁺¹x ]/(m+n) + [m-1] / [m+n] ∫sin^m-2cosⁿx dx