Definite integral question

Question

Calculate the integral of: 

cos [2x] π. cos πx dx

From 0 to 1.

Answer 1

Assuming that the integrand is  cos(2 pi x) cos(pi x) , the integral can be worked by using a double angle formula for the first term  

  cos( 2pi x) =  1 - 2 sin²(pi x)

 

  Working with an indefinite integral approach

 

  ∫ ( 1- 2 sin²(pi x) ) cos( pi x) dx   =

 

  ∫ cos(pi x) dx  - 2 ∫ sin²(pi x) cos(pi x) dx  =

 

   sin(pi x) / pi  - (2/3pi) sin³(pi x)     + C

 

  Since sin(pi x)  is zero for both  x = 1   and x = 0

    the definite integral is zero.