Evalute the definite integral below.

Question

Evalute the definite integral below.

 

 

∫0^pi/2 x cos 2x dx

Answer 1

To find an antiderivative for xcos2x dx use Integration by Parts.

 

For example:

let u = x  and  dv = cos2x dx

   du = dx   and v = 1/2 sin 2x

 

Then the original integral using uv - ∫ vdu becomes:

 

1/2 xsin2x - ∫ 1/2 sin2x dx

 

(1/2 xsin2x +1/4 cos 2x) ]₀^π/2

 

[-1/4] - [1/4] = -1/2

 

See: https://www.desmos.com/calculator/bihvueus7d

 

The negative result comes from the fact that there is more area below the x-axis than above.