JEE Mains · Maths · STD 12 - 7.2 definite integral
If \( \int_{0}^{1}4~cot^{-1}(1-2x+4x^{2})dx=a~tan^{-1}(2)-b~log_{c}(5), \) where a, b \( \in N \), then \( (2a+b) \) is equal to :
- A 7
- B 8
- C 9
- D 10
Answer & Solution
Correct Answer
(C) 9
Step-by-step Solution
Detailed explanation
Let \( I=\int_{0}^{1}cot^{-1}(1-2x+4x^{2})dx \) \( I=\int_{0}^{1}(cot^{-1}(2x-1)-cot^{-1}(2x))dx \)....(1) Applying king \( I=\int_{0}^{1}(-cot^{-1}(2x-1)+cot^{-1}(2x-2))dx \)....(2) From (1) & (2) \( 2I=\int_{0}^{1}(cot^{-1}(2x-2)-cot^{-1}(2x))dx \)…
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
- If \(f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2} < x < \frac{\pi}{2},\) and \(f(0)=0,\) then \(f(1)\) is equal toJEE Mains 2020 Hard
- Let \(\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in R\). If \(x(\alpha, 1,2)+y(1, \beta, 2)\) \(+z(2,3, \gamma)=(0,0,0)\) for some \(x, y, z \in R, x y z \neq\) 0 , then \(6 \alpha+4 \beta+\gamma\) is equal to ..............JEE Mains 2024 Hard
- The area of the region \(A\,\{ \,(x,y)\,\,:\,\,0\,\, \le \,y\, \le \,x\,\left| x \right|\, + \,1\) and \( - \,1\, \le \,x\, \le \,1\,\} \) in sq. units, isJEE Mains 2019 Hard
- Consider the system of linear equations in \(x, y, z\):
\(x + 2y + tz = 0\),
\(6x + y + 5tz = 0\),
\(3x + t^2 y + f(t) z = 0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)JEE Mains 2026 Hard - Let \(S=\left\{ w _1, w _2, \ldots.\right\}\) be the sample space associated to a random experiment. Let \(P \left( w _{ n }\right)=\frac{ P \left( w _{ n -1}\right)}{2}, n \geq 2\).Let \(A=\{2 k +3 \ell ; k , \ell \in N \}\) and \(B=\left\{ W _{ n } ; n \in A \right\}\).Then \(P ( B )\) is equal toJEE Mains 2023 Hard
- \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {1 - \cos \,2x} \right)}^2}}}{{2x\,\tan \,x - x\,\tan \,2x}}\) isJEE Mains 2016 Hard
More PYQs from JEE Mains
- Let \(p\) and \(q\) be two real numbers such that \(p+q=\) 3 and \(p^{4}+q^{4}=369\). Then \(\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}\) is equal toJEE Mains 2022 Hard
- \(ABC\) is a triangular park with \(AB = AC = 100\) \(metres\). A vertical tower is situated at the mid-point of \(BC\). If the angles of elevation of the top of the tower at \(A\) and \(B\) are \({\cot ^{ - 1}}\left( {3\sqrt 2 } \right)\) and \(\cos e{c^{ - 1}}\left( {2\sqrt 2 } \right)\) respectively, then the height of the tower (in metres) isJEE Mains 2019 Hard
- Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}\), \(\overrightarrow{ b }=(1- t ) \hat{ i }+(1+ t ) \hat{ j }+2 \hat{ k }\) and \(\overrightarrow{ c }=\hat{ i }- t \hat{ j }+\hat{ k }, t \in R\) be such that for \(\alpha, \beta, \gamma \in R , \alpha \overrightarrow{ a }+\beta \overrightarrow{ b }+\gamma \overrightarrow{ c }=\overrightarrow{0}\) \(\Rightarrow \alpha=\beta=\gamma=0\). Then, the set of all values of \(t\) is.JEE Mains 2022 Hard
- Let the normal at the point \(P\) on the parabola \(y ^{2}=\) \(6 x\) pass through the point \((5,-8)\). If the tangent at \(P\) to the parabola intersects its directrix at the point \(Q\), then the ordinate of the point \(Q\) isJEE Mains 2022 Medium
- If \(5, 5r, 5r^2\) are the lengths of the sides of a triangle, then \(r\) cannot be equal toJEE Mains 2019 Hard
- Given sum of the first \(n\) terms of an \(A.P.\) is \(2n + 3n^2.\) Another \(A.P.\) is formed with the same first term and double of the common difference, the sum of \(n\) terms of the new \(A.P.\) isJEE Mains 2013 Hard