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Question Number 137940 by benjo_mathlover last updated on 08/Apr/21

∫_1 ^( e) (√(1+(1/x^2 )+(1/((x+1)^(2 ) )))) dx =?

∫1e1+1x2+1(x+1)2dx=?

Answered by john_santu last updated on 08/Apr/21

J = ∫_1 ^( e) (√(((x^2 +1)/x^2 )+(1/((x+1)^2 )))) dx J= ∫_1 ^( e) (√(((x^2 +1)(x+1)^2 +x^2 )/(x^2 (x+1)^2 ))) dx J= ∫_1 ^( e) (√(((x^2 +x+1)^2 )/(x^2 (x+1)^2 ))) dx J=∫_1 ^( e) ((x^2 +x+1)/(x(x+1))) dx J= ∫_1 ^( e) (1+(1/x)−(1/(x+1)))dx = [ x +ln ∣x∣−ln ∣x+1∣ ]_1 ^e = (e+1−ln (e+1))−(1−ln 2) =e+ln ((2/(e+1)))

J=∫1ex2+1x2+1(x+1)2dxJ=∫1e(x2+1)(x+1)2+x2x2(x+1)2dxJ=∫1e(x2+x+1)2x2(x+1)2dxJ=∫1ex2+x+1x(x+1)dxJ=∫1e(1+1x−1x+1)dx=[x+ln∣x∣−ln∣x+1∣]1e=(e+1−ln(e+1))−(1−ln2)=e+ln(2e+1)

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