Question

How do i solve part (b) of this question?

i was solving my math assignment until i came across part b. i have no idea what the numbers in the piecewise function after the equal is, or how do i solve it.

Answer 1

See below.

Explanation:

The integration limits `{0, pi/2}` make null the term `1/n cos^(n-1)x sin x` then calling

`I_n = int_0^(pi/2) cos^n x dx` we have

`I_n = (n-1)/n I_(n-2)` and for

`n` odd

`I_(2n-1) = (2n-2)/(2n-1)I_(2n-3)` with `I_1 = int_0^(pi/2) cos x dx = 1`

`n` even

`I_(2n) = (2n-1)/(2n)I_(2n-2)` with `I_0 = int_0^(pi/2) cos^0 x dx = pi/2`

and then for `n` odd

`I_3 = 2/3I_1 = 2/3`
`I_5 = 4/5I_3 = (2/3)(4/5)`
`cdots`
`I_(2n-1)=(2/3)(4/5) cdots(2n-2)/(2n-1)`

analogously for `n` even

`I_2 = 1/2 I_0 =(pi/2)(1/2)`
`I_4 = 3/4 I_2 = (pi/2)(1/2)(3/4)`

`cdots`

`I_(2n) = (pi/2)(1/2)(3/4) cdots (2n-1)/(2n)`