
Find the value of the definite integral \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\].
A. \[ - 1\]
B. 2
C. \[1 + {e^{ - 1}}\]
D. None of these
Answer
163.8k+ views
Hint: Here, a definite integral is given. First, use the sum rule of the integration \[\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]} dx = \int\limits_a^b {f\left( x \right)} dx + \int\limits_a^b {g\left( x \right)} dx\] and split the terms of the given integral. Then, solve the integrals by applying the integration formulas.
Formula Used:Integration Formula: \[\int\limits_a^b {ndx = \left[ {nx} \right]} _a^b = n\left( {b - a} \right)\]
The sum rule of the integration: \[\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]} dx = \int\limits_a^b {f\left( x \right)} dx + \int\limits_a^b {g\left( x \right)} dx\]
Complete step by step solution:The given definite integral is \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\].
Let consider,
\[I = \int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\]
Apply the sum rule of the integration \[\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]} dx = \int\limits_a^b {f\left( x \right)} dx + \int\limits_a^b {g\left( x \right)} dx\].
\[I = \int\limits_0^1 1 dx + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
Solve the first integral by applying the formula \[\int\limits_a^b {ndx = \left[ {nx} \right]} _a^b = n\left( {b - a} \right)\].
\[I = \left[ x \right]_0^1 + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
\[ \Rightarrow I = \left( {1 - 0} \right) + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
\[ \Rightarrow I = 1 + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
But the second term \[\int\limits_0^1 {{e^{ - {x^2}}}} dx\] is not integrable.
Therefore, the value of the integral \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\] does not exists.
Option ‘D’ is correct
Note: Students often get confused and solve the second integral \[\int\limits_0^1 {{e^{ - {x^2}}}} dx\] by applying the integration formula \[\int {{e^x}} dx = {e^x}\] . Because of that, they get the value of the integrals as \[\int\limits_0^1 {{e^{ - {x^2}}}} dx = \dfrac{1}{e} - 1\] and \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx = 1 + \dfrac{1}{e} - 1 = \dfrac{1}{e}\]. But both values are wrong, so they get the wrong solution.
Formula Used:Integration Formula: \[\int\limits_a^b {ndx = \left[ {nx} \right]} _a^b = n\left( {b - a} \right)\]
The sum rule of the integration: \[\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]} dx = \int\limits_a^b {f\left( x \right)} dx + \int\limits_a^b {g\left( x \right)} dx\]
Complete step by step solution:The given definite integral is \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\].
Let consider,
\[I = \int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\]
Apply the sum rule of the integration \[\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]} dx = \int\limits_a^b {f\left( x \right)} dx + \int\limits_a^b {g\left( x \right)} dx\].
\[I = \int\limits_0^1 1 dx + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
Solve the first integral by applying the formula \[\int\limits_a^b {ndx = \left[ {nx} \right]} _a^b = n\left( {b - a} \right)\].
\[I = \left[ x \right]_0^1 + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
\[ \Rightarrow I = \left( {1 - 0} \right) + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
\[ \Rightarrow I = 1 + \int\limits_0^1 {{e^{ - {x^2}}}} dx\]
But the second term \[\int\limits_0^1 {{e^{ - {x^2}}}} dx\] is not integrable.
Therefore, the value of the integral \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx\] does not exists.
Option ‘D’ is correct
Note: Students often get confused and solve the second integral \[\int\limits_0^1 {{e^{ - {x^2}}}} dx\] by applying the integration formula \[\int {{e^x}} dx = {e^x}\] . Because of that, they get the value of the integrals as \[\int\limits_0^1 {{e^{ - {x^2}}}} dx = \dfrac{1}{e} - 1\] and \[\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} dx = 1 + \dfrac{1}{e} - 1 = \dfrac{1}{e}\]. But both values are wrong, so 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 Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

JEE Mains 2025 Cut-Off GFIT: Check All Rounds Cutoff Ranks

Lami's Theorem

Other Pages
Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks

NEET Marks vs Rank 2024|How to Calculate?

NEET 2025: All Major Changes in Application Process, Pattern and More
