
What is the value of the definite integral \[\int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx\] ?
A. \[{e^{1000}} - 1\]
B. \[\dfrac{{{e^{1000}} - 1}}{{e - 1}}\]
C. \[1000\left( {e - 1} \right)\]
D. \[\dfrac{{e - 1}}{{1000}}\]
Answer
160.8k+ views
Hint: Here, a definite integral is given. First, simplify the integral by applying the integration rule \[\int\limits_0^{na} {f\left( x \right) dx} = n\int\limits_0^a {f\left( x \right) dx} \]. Then, calculate the value of \[\left[ x \right]\] in the new obtained interval. After that, solve the integral by using the integration formula \[\int {{e^x}} dx = {e^x}\] . In the end, apply the limits and solve the equation to get the required answer.
Formula Used:\[\int\limits_0^{na} {f\left( x \right) dx} = n\int\limits_0^a {f\left( x \right) dx} \]
\[\int {{e^x}} dx = {e^x}\]
Greatest integer function: \[\left[ x \right] = n\], where \[n \le x < n + 1\]
Complete step by step solution:The given definite integral is \[\int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx\].
Let consider,
\[I = \int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx\]
Apply the integration rule \[\int\limits_0^{na} {f\left( x \right) dx} = n\int\limits_0^a {f\left( x \right) dx} \] on the right-hand side.
\[I = 1000\int\limits_0^1 {{e^{x - \left[ x \right]}}} dx\]
Now calculate the value of \[\left[ x \right]\] in the interval 0 to 1.
Apply the greatest integer function \[\left[ x \right] = n\], where \[n \le x < n + 1\].
We get,
In the interval \[0 \le x < 1\]
\[\left[ x \right] = 0\]
Thus,
\[I = 1000\int\limits_0^1 {{e^{x - 0}}} dx\]
\[ \Rightarrow I = 1000\int\limits_0^1 {{e^x}} dx\]
Solve the integral by using the integration formula \[\int {{e^x}} dx = {e^x}\].
\[ \Rightarrow I = 1000\left[ {{e^x}} \right]_0^1\]
Apply the upper and lower limits.
\[ \Rightarrow I = 1000\left( {{e^1} - {e^0}} \right)\]
\[ \Rightarrow I = 1000\left( {e - 1} \right)\]
Therefore, \[\int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx = 1000\left( {e - 1} \right)\].
Option ‘C’ is correct
Note: Students get confused and directly use the integration formula \[\int {{e^x}} dx = {e^x}\] to solve the integral and after that they solve for the greatest value function. Because of that, they get the wrong solution.
Formula Used:\[\int\limits_0^{na} {f\left( x \right) dx} = n\int\limits_0^a {f\left( x \right) dx} \]
\[\int {{e^x}} dx = {e^x}\]
Greatest integer function: \[\left[ x \right] = n\], where \[n \le x < n + 1\]
Complete step by step solution:The given definite integral is \[\int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx\].
Let consider,
\[I = \int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx\]
Apply the integration rule \[\int\limits_0^{na} {f\left( x \right) dx} = n\int\limits_0^a {f\left( x \right) dx} \] on the right-hand side.
\[I = 1000\int\limits_0^1 {{e^{x - \left[ x \right]}}} dx\]
Now calculate the value of \[\left[ x \right]\] in the interval 0 to 1.
Apply the greatest integer function \[\left[ x \right] = n\], where \[n \le x < n + 1\].
We get,
In the interval \[0 \le x < 1\]
\[\left[ x \right] = 0\]
Thus,
\[I = 1000\int\limits_0^1 {{e^{x - 0}}} dx\]
\[ \Rightarrow I = 1000\int\limits_0^1 {{e^x}} dx\]
Solve the integral by using the integration formula \[\int {{e^x}} dx = {e^x}\].
\[ \Rightarrow I = 1000\left[ {{e^x}} \right]_0^1\]
Apply the upper and lower limits.
\[ \Rightarrow I = 1000\left( {{e^1} - {e^0}} \right)\]
\[ \Rightarrow I = 1000\left( {e - 1} \right)\]
Therefore, \[\int\limits_0^{1000} {{e^{x - \left[ x \right]}}} dx = 1000\left( {e - 1} \right)\].
Option ‘C’ is correct
Note: Students get confused and directly use the integration formula \[\int {{e^x}} dx = {e^x}\] to solve the integral and after that they solve for the greatest value function. Because of that, they get the wrong solution.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2026 Syllabus PDF - Download Paper 1 and 2 Syllabus by NTA

JEE Main Eligibility Criteria 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

JEE Advanced 2025 Notes
