JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(f\left( x \right) = \frac{{2 - x\,\cos \,x}}{{2 + x\,\cos \,x}}\) and \(g\left( x \right) = {\log _e}\,x\), \(\left( {x > 0} \right)\) then the value of the integral \(\int\limits_{\frac{{ - \pi }}{4}}^{\frac{\pi }{4}} {g\left( {f\left( x \right)} \right)} dx\) is
- A \({\log _e}\,1\)
- B \({\log _e}\,2\)
- C \({\log _e}\,e\)
- D \({\log _e}\,3\)
Answer & Solution
Correct Answer
(A) \({\log _e}\,1\)
Step-by-step Solution
Detailed explanation
\(g(f(x))=\ln (f(x))=\ln \left(\frac{2-x \cdot \cos x}{2+x \cdot \cos x}\right)\) \(\therefore \quad \mathrm{I}=\int_{0}^{\pi / 4}\left(\ln \left(\frac{2-x \cdot \cos x}{2+x \cdot \cos x}\right)+\ell\left(\frac{2+x \cdot \cos x}{2-x \cdot \cos x}\right)\right) d x\)…
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
- Let the circumcentre of a triangle with vertices \(A ( a , 3), B ( b , 5)\) and \(C ( a , b ), ab >0\) be \(P (1,1)\). If the line \(AP\) intersects the line \(BC\) at the point \(Q \left( k _{1}, k _{2}\right)\), then \(k _{1}+ k _{2}\) is equal to.JEE Mains 2022 Hard
- Consider a circle \(C_1: x^2+y^2-4 x-2 y=\alpha-5\).Let its mirror image in the line \(y=2 x+1\) be another circle \(C _2: 5 x ^2+5 y ^2-10 fx -10 gy +36=0\).Let \(r\) be the radius of \(C _2\). Then \(\alpha+ r\) is equal to \(......\).JEE Mains 2023 Hard
- In a certain town, \(25\%\) of the families own a phone and \(15\%\) own a car; \(65\%\) families own neither a phone nor a car and \(2,000\) families own both a car and a phone. Consider the following three statements \((A)\,\,\,5\%\) families own both a car and a phone
\((B)\,\,\,35\%\) families own either a car or a phone
\((C)\,\,\,40,000\) families live in the town
Then,JEE Mains 2015 Hard - A variable \(X\) takes values \(0, 0, 2, 6, 12, 20, \ldots, n(n-1)\) with frequencies \({}^nC_0, {}^nC_1, {}^nC_2, {}^nC_3, {}^nC_4, {}^nC_5, \ldots, {}^nC_n\), respectively. If the mean of this data is \(60\), then its median is :JEE Mains 2026 Hard
- Consider the line \(L\) given by the equation \(\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}\). Let \(Q\) be the mirror image of the point \((2,3,-1)\) with respect to \(L\). Let a plane \(P\) be such that it passes through \(Q\), and the line \(L\) is perpendicular to \(P.\) Then which of the following points is on the plane \(P\) ?JEE Mains 2021 Medium
- Let \(n \geq 5\) be an integer. If \(9^{n}-8 n-1=64 \alpha\) and \(6^{ n }-5 n -1=25 \beta\), then \(\alpha-\beta\) is equal toJEE Mains 2022 Medium
More PYQs from JEE Mains
- The mean and the standard deviation \((s.d.)\) of five observations are \(9\) and \(0,\) respectively. If one of the observations is changed such that the mean of the new set of five observations becomes \(10,\) then their \(s.d.\) is?JEE Mains 2018 Hard
- If \(f\) and \(g\) are differentiable functions in \([0, 1]\) satisfying \(f\left( 0 \right) = 2 = g\left( 1 \right)\;,\;\;g\left( 0 \right) = 0,\) and \(f\left( 1 \right) = 6,\) then for some \(c \in \left] {0,1} \right[\) . .JEE Mains 2014 Medium
- A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is \(\tan ^{-1} \frac{3}{4}\). Water is poured in it at a constant rate of \(6\) cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is \(4\) meters, is.JEE Mains 2022 Hard
- Let \([t]\) denote the greatest integer less than or equal to \(t.\) Then, the value of the integral \(\int\limits_{0}^{1}\left[-8 x^{2}+6 x-1\right] d x\) is equal toJEE Mains 2022 Hard
- Let for some \(\alpha \in \mathbb{R}\), \(f:\mathbb{R}\rightarrow\mathbb{R}\) be a function satisfying \(f(x+y)=f(x)+2y^2+y+\alpha xy\) for all \(x,y \in \mathbb{R}\). If \(f(0)=-1\) and \(f(1)=2\), then the value of \(\sum_{n=1}^{5}(\alpha+f(n))\) is:JEE Mains 2026 Hard
- Let \(S\) be the set of all real values of \(\lambda \) such that a plane passing through the points \(( - {\lambda ^2},1,1),(1, - {\lambda ^2},1)\) and \((1,1, - {\lambda ^2})\) also passes through the point \((-1, -1, 1).\) Then \(S\) is equal toJEE Mains 2019 Hard