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Question
`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.
Options
log |1 + cosx| + C
log |x + sinx| + C
`x - tan x/2 + "C"`
`x.tan x/2 + "C"`
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Solution
`int (x + sinx)/(1 + cosx) "d"x` is equal to `x.tan x/2 + "C"`.
Explanation:
I = `int (x + sinx)/(1 + cosx) "d"x`
= `int x/(1 + cos x) "d"x + int (sinx)/(1 + cosx) "d"x`
= `int x/(2cos^2 x/2) "d"x + int (2sin x/2 cos x/2)/(2cos^2 x/2) "d"x`
= `int x sec^2 x/2 "d"x + int tan x/2 "d"x`
= `1/2 [x*2 tan x/2 - int 2 tan x/2 "d"x] + int tan x/2 "d"x`
= `x * tan x/2 + "C"`
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