Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • I have a doubt, kindly clarify. - The integral is equal to : - Integral Calculus - JEE Main

The integral \int_{1}^{e}\left \{ \left ( \frac{x}{e} \right )^{2x}-\left ( \frac{e}{x} \right )^{x} \right \}\log_{e}xdx  is equal to : 

  • Option 1)

     

    -\frac{1}{2}+\frac{1}{e}-\frac{1}{2e^{2}}

     

     

     

  • Option 2)

     

    \frac{3}{2}-e-\frac{1}{2e^{2}}

  • Option 3)

     

    \frac{3}{2}-\frac{1}{e}-\frac{1}{2e^{2}}

  • Option 4)

     

    \frac{1}{2}-e-\frac{1}{e^{2}}

Answers (1)

best_answer

 

lower and upper limit -

\int_{a}^{b}f\left ( x \right )dx= \left ( F\left ( x \right ) \right )_{a}^{b}

                = F\left ( b \right )-F\left ( a \right )

 

- wherein

Where a is lower and b is upper limit.

 

 

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

\left ( \frac{x}{e} \right )^{x}=t\Rightarrow x(ln\: x-1)=lnt\\\\(ln\: x)dx=\frac{dt}{t}\\\\\\I=\int_{\frac{1}{e}}^{1}(t^{2}-\frac{1}{t})\frac{dt}{t}=\int_{\frac{1}{e}}^{1}(t^{2}-\frac{1}{t})\frac{dt}{t}=\int_{\frac{1}{e}}^{1}(t-\frac{1}{t^{2}})dt=\left [ \frac{t^{2}}{2} +\frac{1}{t} \right ]_{\frac{1}{e}}^{1}\\\\=(\frac{1}{2}+1)-(\frac{1}{2e^{2}}+e)=\frac{3}{2}-\frac{1}{2e^{2}}-e

 

 


Option 1)

 

-\frac{1}{2}+\frac{1}{e}-\frac{1}{2e^{2}}

 

 

 

Option 2)

 

\frac{3}{2}-e-\frac{1}{2e^{2}}

Option 3)

 

\frac{3}{2}-\frac{1}{e}-\frac{1}{2e^{2}}

Option 4)

 

\frac{1}{2}-e-\frac{1}{e^{2}}

Posted by

admin

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 application correction begins on jeemain.nta.nic.in; edit details by December 2
anam-khan

JEE Mains 2026 LIVE: NTA to conduct session 1 from January 21 to 30; exam pattern, syllabus
anam-khan

JEE Main 2026 correction window opens tomorrow; NTA allows exam city change

Latest Question