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1-e-tan-1-x-x-log-x-2-1-dx-

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Question Number 101816 by Dwaipayan Shikari last updated on 04/Jul/20
∫_1 ^( e) (((tan^(−1) x)/x)+((log)/(x^2 +1)))dx
∫1e(tan−1xx+logx2+1)dx
Answered by Ar Brandon last updated on 05/Jul/20
Let I=∫_1 ^e {((tan^(−1) x)/x)+((logx)/(x^2 +1))}dx (d/dx){tan^(−1) x∙logx}=((tan^(−1) x)/x)+((logx)/(1+x^2 )) ⇒I=∫_1 ^e d(tan^(−1) x∙logx)=[tan^(−1) x∙logx]_1 ^e =tan^(−1) (e)
LetI=∫1e{tan−1xx+logxx2+1}dxddx{tan−1x⋅logx}=tan−1xx+logx1+x2⇒I=∫1ed(tan−1x⋅logx)=[tan−1x⋅logx]1e=tan−1(e)

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