Mathematics

# Evaluate the definite integral :$\displaystyle \int_{e}^{e^2} \left\{\dfrac {1}{\log x} -\dfrac {1}{(\log x)^2}\right\} dx$

##### SOLUTION
$I=\displaystyle \int_e^{e^2}\dfrac {1}{\log x}.1\displaystyle \int_e^{e^2}\dfrac {1}{(\log x)^2}dx$

$I$                       $II$

For the first part of the integral, taking $I$ as the first function and $II$ as the second function

we have,

$\left[\dfrac {x}{\log x}\right]_e^{e^2}-\displaystyle \int_e^{e^2}\dfrac {1}{x(\log x)^2}x\ dx-\displaystyle \int_e^{e^2}\dfrac {1}{(\log x)^2}dx$

$\therefore \ I=\dfrac {e^2}{\log e^2}-\dfrac {e}{\log e}$

$=\dfrac {e^2}{2\log e}-\dfrac {e}{\log e}$

$=\dfrac {e^2}{2}-e$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Medium
The value of $\displaystyle \int \dfrac{dx}{4x^2 +9}$ is:
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Q2 Subjective Medium
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Q3 Subjective Medium
Solve $\int _{ 0 }^{ \pi /2 }{ { e }^{ x }\cos xdx }$.

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Q4 Single Correct Medium
$\displaystyle \int \dfrac { dx }{ x{ \left( 1+\sqrt [ 3 ]{ x } \right) }^{ 2 } }$ is equal to:
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Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.