QJEEM23Ma8cd2: JEE Main 2023 Mathematics - Class 12 JEEM23Ma8cd2 Solution: The value of the integral ∫ − log e ⁡ 2 log e ⁡ 2 e x ( log e ⁡ …

Question

Verified Solution • 2025–26 Session moderate

The value of the integral $\int_{- \log_{e} 2}^{\log_{e} 2} e^{x} (\log_{e} (e^{x} + \sqrt{1 + e^{2 x}})) d x$ 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 $\log _{e}\left(\frac{\sqrt{2}(3-\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$

Accepted Answer

\int_{- \log_{e} 2}^{\log_{e} 2} e^{x} (\log_{e} (e^{x} + \sqrt{1 + e^{2 x}})) d x

Let

e^x=t \Rightarrow e^x d x=d t

When,

x \rightarrow-\log _e 2

, then

t \rightarrow \frac{1}{2}

When,

x \rightarrow \log _e 2

, then

t \rightarrow 2

I = \int_{\frac{1}{2}}^{2} [\log_{e} (t + \sqrt{1 + t^{2}})] d t

...........(i)

On applying integration by part method in Eq. (i), we get

\begin{aligned}&amp; I=\left[t \log _e\left(t+\sqrt{1+t^2}\right)\right]_{1 / 2}^2-\int_{1 / 2}^2 \frac{t}{t+\sqrt{1+t^2}}\left(1+\frac{2 t}{2 \sqrt{1+t^2}}\right) d t \\\\&amp; =2 \log _e(2+\sqrt{5})-\frac{1}{2} \log _e\left(\frac{1+\sqrt{5}}{2}\right)-\int_{1 / 2}^2 \frac{t}{\sqrt{1+t^2}} d t\end{aligned}

=\log _e\left(\frac{(2+\sqrt{5})^2}{\left(\frac{1+\sqrt{5}}{2}\right)^{1 / 2}}\right)-\frac{1}{2} \int_{1 / 2}^2 \frac{2 t}{\sqrt{1+t^2}} d t

.............(ii)

Let

\quad I_1=\int_{1 / 2}^2 \frac{2 t}{\sqrt{1+t^2}} d t

Let

1+t^2=w

2 t d t=d w

When,

t \rightarrow \frac{1}{2}

, then

w=\frac{5}{4}

When,

t \rightarrow 2

, then

w=5

\begin{aligned}I_1 &amp; =\int_{5 / 4}^5 \frac{1}{\sqrt{w}} d w \\\\&amp; =[2 \sqrt{w}]_{5 / 4}^5 \\\\&amp; =2\left[\sqrt{5}-\frac{\sqrt{5}}{2}\right]=\sqrt{5}\end{aligned}

On substitute value of

I_1

in Eq. (ii), we get

I=\log _e\left(\frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}

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