The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:
- \( \sqrt{5} - \sqrt{2} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
- \( \sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
- \( \sqrt{5} - \sqrt{2} + \log_e \left( \frac{7 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
- \( \sqrt{2} - \sqrt{5} + \log_e \left( \frac{7 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
The Correct Option is B
Approach Solution - 1
To solve the integral \(\int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx\), we can use the properties of logarithms and integration by parts. Let's break down the steps:
- \(\text{Consider the substitution } x = \sinh t. \text{ Then, } dx = \cosh t \, dt. \text{ Thus, } \sqrt{x^2 + 1} = \cosh t.\)
By using the hyperbolic identities:
- \(x = \sinh t \Rightarrow \sqrt{x^2 + 1} = \cosh t.\)
- \(\text{Note: } \sinh t + \cosh t = e^t. \text{ Therefore, the integral becomes:}\)
Change the limits from \(x\) to \(t\) conditions:
- \(x = -1 \Rightarrow \sinh t = -1 \Rightarrow t = \text{arcsinh}(-1)\)
- \(x = 2 \Rightarrow \sinh t = 2 \Rightarrow t = \text{arcsinh}(2)\)
Evaluate the integral:
- \(\int t \, dt = \frac{t^2}{2} \bigg|_{\text{arcsinh}(-1)}^{\text{arcsinh}(2)}.\)
Calculate the evaluated boundaries:
- \(\frac{1}{2} \left( (\text{arcsinh}(2))^2 - (\text{arcsinh}(-1))^2 \right).\)
Compute the values using the property \(\text{arcsinh}(x) = \log_e(x + \sqrt{x^2 + 1})\):
- \(\text{arcsinh}(2) = \log_e(2 + \sqrt{5}),\ \text{arcsinh}(-1) = \log_e(-1 + \sqrt{2}).\)
Thus:
- \(\frac{1}{2} \left( (\log_e (2 + \sqrt{5}))^2 - (\log_e (-1 + \sqrt{2}))^2 \right).\)
Put the values back to finalize the computation:
- \(\sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right).\)
Thus, the value of the integral is \(\sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right).\)
Approach Solution -2
Let:
\[ f(x) = \log_e \left( x + \sqrt{x^2 + 1} \right) .\]
Use substitution to simplify the integral. Define \( u = x + \sqrt{x^2 + 1} \), so:
\[ du = \left( 1 + \frac{x}{\sqrt{x^2 + 1}} \right) dx = \frac{\sqrt{x^2 + 1} + x}{\sqrt{x^2 + 1}} dx = \frac{u}{\sqrt{x^2 + 1}} dx. \]
Squaring \( u \), we find:
\[ u^2 = x^2 + 1 + 2x \sqrt{x^2 + 1}. \]
Rearrange:
\[ x \sqrt{x^2 + 1} = \frac{u^2 - x^2 - 1}{2}. \]
From symmetry and the bounds \( x \in [-1, 2] \), evaluate \( u \) at \( x = -1 \) and \( x = 2 \):
At \( x = -1 \), \( u = -1 + \sqrt{2} \).
At \( x = 2 \), \( u = 2 + \sqrt{5} \).
Substitute back into the integral and compute:
\[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) dx = \sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right). \]
Final Answer:
\[ \boxed{\sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right)} \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Definite Integral Questions
- The value of \(\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \frac{n^3}{{(n^2 + k^2)(n^2 + 3k^2)}}\) is
The value \( 9 \int_{0}^{9} \left\lfloor \frac{10x}{x+1} \right\rfloor \, dx \), where \( \left\lfloor t \right\rfloor \) denotes the greatest integer less than or equal to \( t \), is ________.
- For \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), if\[y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} \, dx\]and\[\lim_{x \to -\frac{\pi}{2}} y(x) = 0\]then \( y\left(\frac{\pi}{4}\right) \) is equal to
- Evaluate the limit : \(\lim_{x \to \frac{\pi}{2}} \left( \frac{1}{\left( x - \frac{\pi}{2} \right)^3} \int_{\frac{\pi}{2}}^x \cos \left( \frac{1}{t^3} \right) \, dt \right)\)
If the value of the integral
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{\sin^x 2023}} \right) dx = \frac{\pi}{4} (\pi + a) - 2, \]
then the value of \(a\) is:
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,