Question

Evaluate the following using properties of definite integral:

`int_(- pi/2)^(pi/2) sin^2theta  "d"theta`

Answer 1

Let f(θ)= sin²θ

f(– θ) = sin²(– θ)

= [sin (– θ)]²

= [– sin θ]²

= sin²θ

f(– θ) = f(θ)

∴ f(θ) is an even function

`int_(- pi/2)^(pi/2) sin^2theta  "d"theta = 2 xx int_0^(pi/2) sin^2theta  "d"theta`

= `2 xx int_0^(pi/2) ((1 - cos 2theta)/2)  "d"theta`

= `2 xx 1/2 int_0^(pi/2) (1 - cos 2theta)  "d"theta`

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

= `[pi/2 - (sin2(pi/2))/2] - [0]`

= `pi/2 - 0`

= `pi/2`