Question

How do you find the definite integral for: `sin^(2) (5x) * cos^(2) (5x) dx` for the intervals `[8, 11]`?

Answer 1

`~~0.3758`

Explanation:

We use the double angle formula for the sine function

`sin(2A) = 2 sin A cos A`

to write

`sin^(2) (5x) * cos^(2) (5x) = (sin(5x) * cos (5x))^2 = (1/2 sin (10x))^2 = 1/4 sin^2(10x)`

We make use of the double angle formula for the cosine

`cos(2A) = cos^2 A-sin^2 A = 1-2sin^2 A`

to write the integrand in the form

`1/4 (1/2[1-cos(20x)]) = 1/8(1-cos(20x))`

So, finally

`int_8^11 sin^(2) (5x) * cos^(2) (5x) dx = int_8^11 1/8(1-cos(20x)) dx`
`qquad = 1/8 (x-1/20 sin(20x))_8^11`
`qquad = 1/8((11-8)-(sin(220)-sin(160))/20) ~~ 0.3758`