Question

`int_{pi/6}^{pi/3} sin^2x dx` = ______ 

Options

• `pi/6`

• `pi/3`

• `pi/12`

• `pi/24`

Answer 1

`int_{pi/6}^{pi/3} sin^2x dx` = `underline(pi/12)`

Explanation:

Let I = `int_{pi/6}^{pi/3} sin^2x dx` .............(i)

Let I = `int_{pi/6}^{pi/3} sin^2(pi/2 - x)dx` .......`[∵ int_a^b f(x)dx = int_a^b f(a + b - x)dx]`

∴ `I = int_{pi/6}^{pi/3} cos^2x dx` .............(ii)

Adding (i) and (ii), we get

2I = `int_{pi/6}^{pi/3} 1 . dx = [x]_{pi/6}^{pi/3}`

= `pi/3 - pi/6 = pi/6`

∴ I = `pi/12`