JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(2\int_0^1 {{{\tan }^{ - 1}}}\,xdx = \int_0^1 {{{\cot }^{ - 1}}}\,(1 - x + {x^2})dx,\) then \(\int_0^1 {{{\tan }^{ - 1}}}\, (1 - x + {x^2})dx\) is equal to
- A \(\frac{\pi }{2} + \log \,2\)
- B \(\log \,2\)
- C \(\frac{\pi }{2} - \log \,4\)
- D \(\log \,4\)
Answer & Solution
Correct Answer
(B) \(\log \,2\)
Step-by-step Solution
Detailed explanation
\(2 \int_{0}^{1} \tan ^{-1} x d x=\int_{0}^{1}\left(\frac{\pi}{2}-\tan ^{-1}\left(1-x+x^{2}\right)\right) d x\) \(2 \int_{0}^{1} \tan ^{-1} x d x=\int_{0}^{1} \frac{\pi}{2} d x-\int_{0}^{1} \tan ^{-1}\left(1-x+x^{2}\right) d x\)…
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
- If \(\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx\), then \(\alpha^2\) is equal to _______.JEE Mains 2026 Hard
- If three positive numbers \(a, b\) and \(c\) are in \(A.P.\) such that \(abc\, = 8\), then the minimum possible value of \(b\) isJEE Mains 2017 Hard
- Let \(a,b \in R,\left( {a \ne 0} \right)\). if the function \(f\) defined as \(f\left( x \right)\left\{ \begin{array}{l}
\frac{{2{x^2}}}{a}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,0 \le x < 1\,\,\,\\
a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,1 \le x < \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
\frac{{2{b^2} - 4b}}{{{x^3}}}\,\,\,,\,\,\,\,\,\sqrt 2 \le x < \infty
\end{array} \right.\,\,\,\,\) is continuous in the interval \(\left[ {0,\infty } \right)\) , then an ordered pair \((a, b)\) isJEE Mains 2016 Hard - Consider the lines \(\mathrm{x}(3 \lambda+1)+\mathrm{y}(7 \lambda+2)=17 \lambda+5\), \(\lambda\) being a parameter, all passing through a point P . One of these lines (say L) is farthest from the origin. If the distance of \(L\) from the point \((3,6)\) is \(d\), then the value of \(d^2\) isJEE Mains 2025 Easy
- Let \(m\) and \(M\) be respectively the minimum and maximum values of \(\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|\). Then the ordered pair \(( m , M )\) is equal toJEE Mains 2020 Hard
- Let the point, on the line passing through the points \(P(1,-2,3)\) and \(Q(5,-4,7)\), farther from the origin and at a distance of \(9\) units from the point \(\mathrm{P}\), be \((\alpha, \beta, \gamma)\). Then \(\alpha^2+\beta^2+\gamma^2\) is equal to :JEE Mains 2024 Medium
More PYQs from JEE Mains
- Let \(O \) be the vertex and \(Q\) be any point on the parabola \({x^2} = 8y\) .If the point \(P\) divides the line segment \(OQ\) internally in the ratio \( 1:3\) , then locus of \(P\) is :JEE Mains 2015 Hard
- In the expansion of \(\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}\), \(x>0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:JEE Mains 2026 Medium
- Let the point A divide the line segment joining the points \(P(-1,-1,2)\) and \(Q(5,5,10)\) internally in the ratio \(\mathrm{r}: 1(\mathrm{r}\gt0)\). If O is the origin and \((\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^2=10\), then the value of r is :JEE Mains 2025 Easy
- If \(S=\{z \in C:|z-i|=|z+i|=|z-1|\}\), then, \(n(S)\) is:JEE Mains 2024 Medium
- Let \(f : R \to R\) be defined by \(f\left( x \right) = \frac{{\left| x \right| - 1}}{{\left| x \right| + 1}}\) then \(f\) isJEE Mains 2014 Hard
- If \(\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\dfrac{\pi}{3}\right)\cot\left(x+\dfrac{\pi}{3}\right)+1\right)dx = \alpha\log_e(\sqrt{3}-1)\), then \(9\alpha^2\) is equal to ________.JEE Mains 2026 Hard