enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 7.2 definite integral
Consider the integral \(I=\int_{0}^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x,\) where \([ x ]\) denotes the greatest integer less than or equal to \(x\). Then the value of \(I\) is equal to:
- A \(9( e -1)\)
- B \(45( e +1)\)
- C \(45( e -1)\)
- D \(9( e +1)\)
Answer & Solution
Correct Answer
(C) \(45( e -1)\)
Step-by-step Solution
Detailed explanation
\(I =\int_{0}^{10}[ x ] \cdot e ^{[ x ]- x +1}\) \(I =\int_{0}^{1} 0 d x +\int_{1}^{2} 1 \cdot e ^{2- x }+\int_{2}^{3} 2 \cdot e ^{3- x }+\ldots . .+\int_{9}^{10} 9 \cdot e ^{10- x } dx\) \(\Rightarrow I=\sum_{n=0}^{9} \int_{n}^{n+1} n \cdot e^{n+1-x} d x\)…
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
- If both the means and the standard deviation of \(50\) observations \(x_1, x_2, ………, x_{50}\) are equal to \(16\) , then the mean of \((x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2\) isJEE Mains 2019 Hard
- The set of all values of \(a\) for which \(\operatorname{Lim}_{x \rightarrow a}([x-5]-[2 x+2])=0\), where \([\propto]\) denotes the greater integer less than or equal to \(\propto\) is equal toJEE Mains 2023 Hard
- Let the eccentricity of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) be \(\frac{5}{4}\). If the equation of the normal at the point \(\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)\) on the hyperbola is \(8 \sqrt{5} x +\beta y =\lambda\), then \(\lambda-\beta\) is equal toJEE Mains 2022 Medium
- If the coefficient of \(x ^{10}\) in the binomial expansion of \(\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}\) is \(5^{ k } l\), where \(l, k \in N\) and \(l\) is coprime to \(5\) , then \(k\) is equal toJEE Mains 2022 Hard
- The number of real roots of the equation \(x | x |-5| x +2|+6=0\), isJEE Mains 2023 Hard
- \(\lim _{x \rightarrow 0} \operatorname{cosec} x\)\(\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)\) is:JEE Mains 2025 Medium
More PYQs from JEE Mains
- The remainder when \(3^{2022}\) is divided by \(5\) isJEE Mains 2022 Hard
- If \(\alpha \) and \(\beta \) are the roots of the equation \(375x^2 -25x -2 = 0\), then \(\mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{r = 1}^n {{\alpha ^r}} + \mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{r = 1}^n {{\beta ^r}} \) is equal toJEE Mains 2019 Hard
- \(\int_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x\) is equal toJEE Mains 2023 Medium
- The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal toJEE Mains 2022 Medium
- The urns \(A, B\) and \(C\) contain \(4\) red, \(6\) black;\(5\) red,\(5\) black and \(\lambda\) red,\(4\) black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn \(C\) is \(0.4\) then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola \(y^2=\lambda x\) with one vertex at the vertex of the parabola isJEE Mains 2023 Hard
- \({ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}\) if and only if :JEE Mains 2024 Hard