Question

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

Answer 1

We use the property,

`int_0^a f(t)*dt = int_0^a f(a - t)*dt`

∴ `int_0^1 t^2 sqrtt(1 - t)*dt = int_0^1 (1 - t)^2 sqrt(1 - 1 + t)*dt`

= `int_0^1 (1 - 2t + t^2)sqrt(t)*dt`

= `int_0^1 (t^(1/2) - 2t^(3/2) + t^(5/2))*dt`

= `[(t^(3/2))/(3/2) - 2*(t(5)/(2))/(5/2) + (t^(7/2))/(7/2)]_0^1`

= `(2)/(3)(1)^(3/2) - (4)/(5)(1)^(5/2) + (2)/(7)(1)^(7/2) - 0`

= `(2)/(3) - (4)/(5) + (2)/(7) - 0`

= `(70 - 84 + 30)/(105)`

= `(16)/(105)`.