JEE Main 2023 — Definite Integration Question with Solution
From: JEE Main 2023 (Online) 11th April Morning Shift
Question
The value of the integral \int_\limits{-\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 :
Choose an option
Show full solutionCorrect option: B
Correct answer
B
Step-by-step explanation
\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x
Let
When, , then
When, , then
I=\int_\limits{\frac{1}{2}}^2\left[\log _e\left(t+\sqrt{1+t^2}\right)\right] d t ...........(i)
On applying integration by part method in Eq. (i), we get
.............(ii)
Let
Let
When, , then
When, , then
On substitute value of in Eq. (ii), we get
Let
When, , then
When, , then
I=\int_\limits{\frac{1}{2}}^2\left[\log _e\left(t+\sqrt{1+t^2}\right)\right] d t ...........(i)
On applying integration by part method in Eq. (i), we get
.............(ii)
Let
Let
When, , then
When, , then
On substitute value of in Eq. (ii), we get
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This is a previous-year question from JEE Main 2023, covering the Definite Integration chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.