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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`
Solution
`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`
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