Question

Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`

Answer 1

Let I = `int_0^1 t^5 sqrt(1 - t^2)*dt`

Put t = sin θ
∴ dt = cos θ dθ

When t = 1, θ = sin^–11 = `pi/(2)`

When t = 0, θ = sin^–10 = 0

∴ I = `int_0^(pi/2) sin^5 theta sqrt(1 - sin^2 theta)cos theta*d theta`

I = `int_0^(pi/2) sin^5 theta*cos theta* cos theta*d theta`

= `int_0^(pi/2) sin^5 theta(1 - sin^2 theta)*d theta`

= `int_0^(pi/2) (sin^5 theta - sin^7 theta)*d theta`

= `int_0^(pi/2) sin^5 theta*d theta - int_0^(pi/2) sin^7 thetad theta`.
Using Reduction formula, we get

I = `4/5*2/3 - 6/7*4/5*2/3`

= `(8)/(15)[1 - 6/7]`

= `(8)/(15) xx (1)/(7)`

= `(8)/(105)`.