Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20252 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Definite Integration

Let [.] denote the greatest integer function. If ∫_0^ e^3 [ 1 e^...

Let [.] denote the greatest integer function. If ∫0e3[1ex−1]dx=α−log⁡e2\int_0^{e^3}\left[\frac{1}{\mathrm{e}^{\mathrm{x}-1}}\right] \mathrm{dx}=\alpha-\log _{\mathrm{e}} 2, then α3\alpha^3 is equal to _______ .

Correct answer

0

Step-by-step solution

f(x)=1ex−1=e1−xf(x)=\frac{1}{e^{x-1}}=e^{1-x}
f(x)=2f(x)=11ex−1=2x=1x=1−ln⁡2\begin{array}{c|c}\mathrm{f}(\mathrm{x})=2 \mathrm{f}(\mathrm{x})=1 \frac{1}{\mathrm{e}^{\mathrm{x}-1}}=2 & \mathrm{x}=1 \\ \mathrm{x}=1-\ln 2 &\end{array}
f(0)=e1=2.71f(0)=e¹=2.71
f(e3)=\left(e^3\right)= e^{1-e³} \in(0,1)<br/><br/>\mathrm{I}=∫01−ℓn22dx+∫1−ℓn211dx+∫1e30dx<br/>=\int_0^{1-\ell \mathrm{n} 2}2 \mathrm{dx}+\int_{1-\ell \mathrm{n} 2}^11 \mathrm{dx}+\int_1^{\mathrm{e}^3}0 \mathrm{dx}<br/>=2(1-\ell \operatorname{n} 2-0)+1(1-1+\ln 2)+0<br/>α−ln⁡2=2−ln⁡2<br/>\alpha-\ln2=2-\ln 2
\alpha=22
\alpha^3=$8
Practice Definite Integration on Quantrex Academy →

More from Definite Integration

The value of the definite integral ∫_ 0 ^ 2 1 3^ (x+3)/(2) \, dx is JEE Advanced 2026The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to: JEE Main 2026If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026Let f: ℝ arrow ℝ be such that f(xy) = f(x)f(y) , for all x, y ∈ ℝ and f(0) ≠ 0 . Let g… JEE Main 2026The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026Let (2^ 1-a + 2^ 1+a ) , f(a) , (3^a + 3^ -a ) be in A.P. and α be the minimum value of… JEE Main 2026Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: JEE Main 2026The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026 Full Definite Integration list Definite Integration JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs