JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\text { If } \int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^{3}}{1+4 \pi^{2}}, \alpha \in R\) where \([x]\) is the greatest integer less than or equal to \(x\), then the value of \(\alpha\) is :
- A \(100(1-e)\)
- B \(200\left(1-\mathrm{e}^{-1}\right)\)
- C \(150\left(e^{-1}-1\right)\)
- D \(50(e-1)\)
Answer & Solution
Correct Answer
(B) \(200\left(1-\mathrm{e}^{-1}\right)\)
Step-by-step Solution
Detailed explanation
\(I=\int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{[x / z\}}} d x=100 \int_{0}^{\pi} \frac{\sin ^{2} x}{e^{x / x}} d x\) \(100 \int_{0}^{\pi} e^{-x / \pi} \frac{(1-\cos 2 x)}{2} d x\) \(=50\left\{\int_{0}^{\pi} e^{-x / \pi} d x-\int_{0}^{\pi} e^{-x / \pi} \cos 2 x d x\right\}\)…
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 the sum of the series \(20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .\) upto \(n ^{ th }\) term is \(488\) and the \(n^{\text {th }}\) term is negative, thenJEE Mains 2020 Hard
- Let a circle \(C\) touch the lines \(L_{1}: 4 x-3 y+K_{1}\) \(=0\) and \(L _{2}: 4 x -3 y + K _{2}=0, K _{1}, K _{2} \in R\). If a line passing through the centre of the circle \(C\) intersects \(L _{1}\) at \((-1,2)\) and \(L _{2}\) at \((3,-6)\), then the equation of the circle \(C\) isJEE Mains 2022 Hard
- Consider the relation R on the set \(\{-2,-1,0,1,2\}\) defined by \((a, b) \in R\) if and only if \(1+ab > 0\). Then, among the statements:
I. The number of elements in R is 17
II. R is an equivalence relationJEE Mains 2026 Medium - Let ABC be a triangle. Consider four points \(p _1, p _2\), \(p _3, p _4\) on the side AB , five points \(p _5, p _6, p _7, p _8, p _9\) on the side BC and four points \(p _{10}, p _{11}, p _{12}, p _{13}\) on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points \(p _1, p _2, \ldots . p _{13}\), is ___ .JEE Mains 2026 Medium
- Let \(\vec{a}=a_i \hat{i}+a_2 \hat{j}+a_3 \hat{k}\) and \(\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\) be two vectors such that \(|\vec{a}|=1 ; \quad \vec{a} \cdot \vec{b}=2\) and \(|\vec{b}|=4\). If \(\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}\), then the angle between \(\vec{b}\) and \(\vec{c}\) is equal to :JEE Mains 2024 Hard
- Let \(f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}\) Then \(f\left(\frac{7 \pi}{12}\right) f "\left(\frac{7 \pi}{12}\right)\) is equal toJEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \(S=\left\{z \in C :\left|\frac{z-6 i}{z-2 i}\right|=1\right.\) and \(\left.\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}\right\}\). Then \(\sum_{z \in s}|z|^2\) is equal toJEE Mains 2026 Medium
- Let the coefficients of third, fourth and fifth terms in the expansion of \(\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,\) be in the ratio \(12: 8: 3 .\) Then the term independent of \(x\) in the expansion, is equal to ...... .JEE Mains 2021 Medium
- Which of the following is true for \(y ( x )\) that satisfies the differential equation \(\frac{d y}{d x}=x y-1+x-y ; y(0)=0\)JEE Mains 2021 Medium
- The \(\operatorname{sum} \sum_{n=1}^{21} \frac{3}{(4 n-1)(4 n+3)}\) is equal toJEE Mains 2022 Medium
- If \(X = \{ {4^n} - 3n - 1:n \in N\} \) and \(Y = \{ 9(n - 1):n \in N\} ,\) then \(X \cup Y\) = . . . . .JEE Mains 2014 Medium
- If \(\operatorname{I}(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n\gt0\), then \(I(9,14)+I(10,13)\) isJEE Mains 2025 Easy