Question

Choose the correct option from the given alternatives : 

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` = ______.

Options

• `(4 - pi)/2`

• `(pi - 4)/2`

• `4 - pi/(2)`

• `(4 + pi)/2`

Answer 1

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` = `bb(underline((4 - pi)/2))`.

Explanation:

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2 = int_0^(pi/2) (1- cos^2x)/(1 + cosx)^2dx` 

=  `int_0^(pi/2) ((1 + cosx)(1-cosx))/(1 + cosx)^2dx` 

=  `int_0^(pi/2) (1-cosx)/(1+cosx)dx` 

= `int_0^(pi/2) (2sin^2  x/2)/(2cos^2  x/2)dx`

= `int_0^(pi/2)tan^2  x/2dx`

= `int_0^(pi/2) (sec^2  x/2-1)dx`

= `(tan  x/2)/(1/2) - x`

= `2[tan  x/2-x]_0^(pi/2)`

= `2[tan  pi/4-pi/2]`

= `2 - pi/2`

= `(4-pi)/2`