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Definite Integration question

2023 · 11 Apr · Shift 1 · Q30

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JEE Main
Mathematics
Definite Integration
2023 · 11 Apr · Shift 1 · Q30

Definite Integration question

2023 · 11 Apr · Shift 1 · Q30

JEE MainMathematicsDefinite IntegrationMCQ+4 / −1

The value of the integral

\int_{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x

is equal to :

A
$\log _{e}\left(\frac{(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$
B
$\log _{e}\left(\frac{\sqrt{2}(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$
C
$\log _{e}\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$
D
$\log _{e}\left(\frac{\sqrt{2}(3-\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$

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