Question

The value of `int_2^(π/2) sin^3x  dx` = ______.

Options

• `1/3`

• `4/3`

• `2/3`

• `8/3`

Answer 1

The value of `int_2^(π/2) sin^3x  dx` = `underlinebb(2/3)`.

Explanation:

I = `int_2^(π/2) sin^3x  dx`

= `int_0^(π/2) sin^2x sinx  dx`

= `int_0^(π/2) (1 - cos^2x) sinx  dx`

Put cos x = t

`\implies` sin x dx = – dt

When x = 0, t = 1

When x = `π/2`, t = 0

∴ I = `int_1^0 (1 - t^2)(-dt)`

= `int_1^0 (1 - t^2)(dt)`

= `[t - t^3/3]_1^0`

= `-[0 - (1 - 1/3)]`

= `2/3`