JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(f(t)=\int_0^\pi \frac{2 x d x}{1-\cos ^2 \sin ^2 x}, 0 < t < \pi\), then the value of \(\int_0^{\frac{\pi}{2}} \frac{\pi^2 d t}{f(t)}\) equals ..........
- A \(3\)
- B \(9\)
- C \(1\)
- D \(7\)
Answer & Solution
Correct Answer
(C) \(1\)
Step-by-step Solution
Detailed explanation
\(f(t)=\int_0^\pi \frac{2 x}{1-\cos ^2 t \sin ^2 x} d x\) ..................(\(1\)) \(=2 \int_0^\pi \frac{(\pi-x) d x}{1-\cos ^2 \sin ^2 x}\) ..................(\(2\)) \( 2 f(t)=2 \int_0^\pi \frac{\pi}{1-\cos ^2 t \sin ^2 x} d x \)…
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 \(f(x)=\int_0^t t\left(t^2-9 t+20\right) d t, 1 \leq x \leq 5\). If the range of \(f\) is \([\alpha, \beta]\), then \(4(\alpha+\beta)\) equals :JEE Mains 2025 Easy
- Let \(z \in C\) with \(Im(z) = 10\) and it satisfies \(\frac{{2z - n}}{{2z + n}} = 2i - 1\) for some natural number \(n\). ThenJEE Mains 2019 Hard
- The value of the integral \(\displaystyle\int_0^\infty \dfrac{\log_e(x)}{x^2 + 4}\,dx\) is:JEE Mains 2026 Hard
- Let \(l_{1}\) be the line in \(xy\)-plane with \(x\) and \(y\) intercepts \(\frac{1}{8}\) and \(\frac{1}{4 \sqrt{2}}\) respectively, and \(l_{2}\) be the line in \(zx\)-plane with \(x\) and \(z\) intercepts \(-\frac{1}{8}\) and \(-\frac{1}{6 \sqrt{3}}\) respectively. If \(d\) is the shortest distance between the line \(l_{1}\) and \(l_{2}\), then \(d ^{-2}\) is equal toJEE Mains 2022 Hard
- In a group of \(100\) persons \(75\) speak English and \(40\) speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is \(\alpha\) and the number of persons who speak only Hindi is \(\beta\), then the eccentricity of the ellipse \(25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2\) is \(.......\)JEE Mains 2023 Medium
- Let \(E\) and \(F\) be two independent events. The probability that both \(E\) and \(F\) happen is \(\frac{1}{12}\) and the probability that neither \(E\) nor \(F\) happens is \(\frac{1}{2}\) , then a value of \(\frac{{P(E)}}{{P\left( F \right)}}\) isJEE Mains 2017 Hard
More PYQs from JEE Mains
- The sum of possible values of \(x\) for \(\tan ^{-1}( x +1)+\cot ^{-1}\left(\frac{1}{ x -1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)\) isJEE Mains 2021 Hard
- Let \(\int x^3 \sin x \mathrm{~d} x=g(x)+C\), where \(C\) is the constant of integration. If \(8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z\), then \(\alpha+\beta-\gamma\) equals :JEE Mains 2025 Medium
- If \({e^y} + xy = e\), the ordered pair \(\left( {\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}}} \right)\) at \(x = 0\) is equal toJEE Mains 2019 Hard
- Suppose \(f ( x )\) is a polynomial of degree four, having critical points at \(-1,0,1\) . If \(T =\{ x \in R \mid f ( x )= f (0)\},\) then the sum of squares of all the elements of \(T\) isJEE Mains 2020 Hard
- The sum \(\sum \limits_{n=1}^{\infty} \frac{2 n^2+3 n+4}{(2 n) !}\) is equal to :JEE Mains 2023 Hard
- \(P\) and \(Q\) are two distinct points on the parabola, \(y^2 = 4x\), with parameters \(t\) and \(t_1\) respectively. If the normal at \(P\) passes through \(Q\), then the minimum value of \(t_1^2\) isJEE Mains 2016 Hard