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General
Easy

Question

The value of the integral not stretchy integral subscript e to the power of negative 1 end exponent end subscript superscript e to the power of 2 end exponent end superscript open vertical bar fraction numerator log subscript e end subscript invisible function application blank x over denominator x end fraction close vertical bar dx is :

  1. 3/2    
  2. 5/2    
  3. 3    
  4. 5    

Hint:

integral subscript a superscript b f x d x space equals space integral subscript a superscript c f x d x space plus space integral subscript c superscript b f x d x

The correct answer is: 5/2


    Given, not stretchy integral subscript e to the power of negative 1 end exponent end subscript superscript e to the power of 2 end exponent end superscript open vertical bar fraction numerator log subscript e end subscript invisible function application blank x over denominator x end fraction close vertical bar
    Let, I = not stretchy integral subscript e to the power of negative 1 end exponent end subscript superscript e to the power of 2 end exponent end superscript open vertical bar fraction numerator log subscript e end subscript invisible function application blank x over denominator x end fraction close vertical bar i.e.  open curly brackets fraction numerator negative log x over denominator x end fraction space comma space e to the power of negative 1 end exponent less or equal than x less or equal than 0 comma space close curly brackets
open curly brackets fraction numerator log x over denominator x end fraction comma 0 space less or equal than x less or equal than 2 close curly brackets
    I space equals space integral subscript e to the power of negative 1 end exponent end subscript superscript e to the power of 0 end superscript fraction numerator negative log x over denominator x end fraction d x space plus space integral subscript e to the power of 0 end subscript superscript e squared end superscript fraction numerator log x over denominator x end fraction d x
P u t comma space log space x space equals space t
T a k i n g space d i f f e r e n t i a t i o n space b o t h space s i d e s comma
1 over x d x space equals space d t
    I space equals space integral negative t d t space plus space integral t d t
I space equals space fraction numerator negative t squared over denominator 2 end fraction space plus space t squared over 2
I space equals space minus 1 half left parenthesis right enclose log x right parenthesis squared end enclose subscript e to the power of negative 1 end exponent end subscript superscript e to the power of 0 end superscript space plus space 1 half right enclose left parenthesis log x right parenthesis squared end enclose subscript e to the power of 0 end subscript superscript e squared end superscript
I space equals space minus 1 half open square brackets left parenthesis log space 0 right parenthesis squared space minus space left parenthesis log space e to the power of negative 1 end exponent right parenthesis squared close square brackets space plus space 1 half open square brackets left parenthesis log space e squared right parenthesis squared space minus space left parenthesis log space e to the power of 0 right parenthesis squared close square brackets space
I space equals negative space 1 half left parenthesis negative 1 right parenthesis space space plus space 1 half open square brackets 2 left parenthesis log space e squared right parenthesis close square brackets
I space equals space 1 half space plus space 1 half left parenthesis 4 right parenthesis
I space equals space 1 half space plus space 2 space equals space 5 over 2

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