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Integral of xsinx/(1-cosx ) over 0 to piEvaluate above integral problem
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 = u0.5sec^2x/2 dx = duand x = vvdx = dvsubstittue in the eqn. (1)and after integrating we get,=> xtan(x/2) + c
Put x=2y integral becomes4*∫0 to pi/2 ycoty dyIntegrate 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)/2Note 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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