JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of the integral \(\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right)\, d x\) is:
- A \(1\)
- B \(0\)
- C \(-1\)
- D \(2\)
Answer & Solution
Correct Answer
(B) \(0\)
Step-by-step Solution
Detailed explanation
Let \(I=\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right) \,d x\) \(\because \log \left(x+\sqrt{x^{2}+1}\right)\) is an odd function \(\therefore \int_{-1}^{1} \ln \left(x+\sqrt{x^{2}+1}\right) \,d x=0\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- For any \(\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)\), the expression \(3\,{\left( {\sin \,\theta - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta \) equalsJEE Mains 2019 Hard
- Let a tangent to the Curve \(9 x^2+16 y^2=144\) intersect the coordinate axes at the points \(A\) and \(B\). Then, the minimum length of the line segment \(A B\) is \(.........\)JEE Mains 2023 Hard
- All the points in the set \(S\, = \left\{ {\frac{{\alpha \, + \,i}}{{\alpha \, - \,i}}\,:\,\alpha \, \in \,R} \right\}\,(i\, = \,\sqrt { - 1} )\) lie on aJEE Mains 2019 Hard
- The shortest distance between the lines \(\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}\) and \(x+y+z+1=0\), \(2 x-y+z+3=0\) isJEE Mains 2020 Hard
- If for some \(\mathrm{m}, \mathrm{n} ; { }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3\) and \({ }^{n-1} P_3:{ }^n P_4=1: 8\), then \({ }^n P_{m+1}+{ }^{n+1} C_m\) is equal toJEE Mains 2024 Hard
- The sum of the first ten terms of an A.P. is \(160\) and the sum of the first two terms of a G.P. is \(8\). If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:JEE Mains 2026 Hard
More PYQs from JEE Mains
- Let the digits \(a, b, c\) be in \(A.P.\) Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in \(A.P.\) at least once. How many such numbers can be formed?JEE Mains 2023 Hard
- Let \(\alpha\) and \(\beta\) be the roots of the equation \(5 x^{2}+6 x-2=0 .\) If \(S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots\) then :JEE Mains 2020 Medium
- \(\smallint \frac{{{{\sin }^2}x{{\cos }^2}x}}{{{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}dx\)JEE Mains 2018 Hard
- Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be two functions defined by \(f(x)=\log _{e}\left(x^{2}+1\right)-e^{-x}+1\) and \(g(x)=\frac{1-2 e^{2 x}}{e^{x}}\). Then, for which of the following range of \(\alpha\), the inequality \(f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)\) holds?JEE Mains 2022 Hard
- If \(x^{3} d y+x y d x=x^{2} d y+2 y d x ; y(2)=e\) and \(x\) \(>1,\) then \(y (4)\) is equal toJEE Mains 2020 Hard
- Let \(A =\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]\) and \(B = A - I\). If \(\omega=\frac{\sqrt{3} i -1}{2}\) then the number of elements in the set \(\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }+(\omega B )^{ n }= A + B \right\}\) is equal to \(..........\)JEE Mains 2022 Hard