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Integral of xsinx/(1-cosx ) over 0 to pi Evaluate above integral problem

Integral of xsinx/(1-cosx ) over 0 to pi
Evaluate above integral problem

Grade:

2 Answers

Arun
25763 Points
3 years ago
Dear Javed
∫(x+sinx)/(1+cosx)dx
 
 => write sinx and cosx in half angle formulaes:
 
 
 
 
 
 
 
 
 
 
 
=> (x + 2sinx/2cosx/2)/(2cos^2x/2)dx => 0.5xsec^2x/2 dx + tanx/2 dx …..........................(1)

 
 
 
 
let tanx/2 = u
0.5sec^2x/2 dx = du
and x = vv
dx = dv
substittue in the eqn. (1)
and after integrating we get,
=> xtan(x/2) + c
Aditya Gupta
2080 Points
3 years ago
Put x=2y integral becomes
4*∫0 to pi/2 ycoty dy
Integrate by parts taking first function as y.
y*lnsiny|0 to pi/2 - ∫0 to pi/2 lnsiny dy
= 0 -0 - (-pi ln2)/2
= (πln2)/2
Note that integral of lnsinx from 0 to pi/2 is -piln2/2 (it's a solved ncert example) san also limit y tends to zero ylnsiny is equal to zero

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