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Question
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
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Solution
Let I = `int [log(1 + cosx) - xtan(x/2)]*dx`
= `int [log(1 + cos.x)*1dx - intxtan (x/2)*dx`
= `[log(1 + cosx)]* int 1dx - int {d/dx [log (1 + cosx)]* int 1dx}*dx - xtan (x/2)*dx`
= `[log (1 + cosx)]*(x) - int 1/(1 + cosx)*(0 - sin x)*xdx - int x tan (x/2)*dx`
= `x*log(1 + cosx) + intx* (sinx)/(1 + cosx)*dx - int xtan (x/2)*dx + c`
= `x*log(1 + cosx) + intx*(2sin(x/2)*cos(x/2))/(2cos^2(x/2)*dx - int xtan (x/2)*dx + c`
= `xlog (1 + cosx) + int x*tan(x/2)*dx - intxtan(x/2)*dx + c`
= x·log(1 + cosx) + c.
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