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Question

Integrate x.cos1x

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Solution

xcos1xdx
puttingx=cosθ
dx=isinθdθ
=cosθ×cos1(cosθ)×sinθdθ
=sinθcosθ×θdθ
=122sinθcosθ.θdθ
=12θ.sin2θdθ
=12[θ×cos2θ21×cos2θ2dθ]
=12[θcos2θ2+12×sin2θ2]+c
=θcos2θ418sin2θ+c
=θ(2cos2θ1)418sin2θ+c
=θ(2cos2θ1)414cosθ1cos2θ+c
=(2x21)cos1x4x41x2+c

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