Question

Evaluate the following integrals using properties of integration:

`int_(- pi/4)^(pi/4) sin^2x  "d"x`

Answer 1

I = `int_(- pi/4)^(pi/4) sin^2x  "d"x`

f(x) = sin²x

f(– x) = sin²(– x) = sin²x

f(x) = f(– x)

f(x) is an even function

`int_(- pi/4)^(pi/4) sin^2x  "d"x = 2 int_0^(pi/4)  sin^2 x  "dx`

`int_(- "a")^"a" f(x)  "d"x = 2int_0^"a"f(x)  "d"x`

= `2 int_0^(pi/4) [(1 - cos 2x)/2] "d"x`

= `2/2 [x - (sin 2x)/2]_0^(pi/4)`

= `pi/4 - 1/2`

I = `(pi - 2)/4`