Mathematics

# Integrate : $\displaystyle\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 2 } } d x$

##### SOLUTION
$\Rightarrow \ \displaystyle \int_{1}^{2}\dfrac {lnx}{x^2}dx$

let $\ln x=t$

$\dfrac {1}{x}=\dfrac {dt}{dx}$

$\dfrac {1}{x}dx=dt$

$I=\displaystyle \int_{1}^{2}\dfrac {lnx}{x^2}dx=\displaystyle \int_{|n|=0}^{ln2}\dfrac {te^t}{e^{2t}}dt=\displaystyle \int_{0}^{ln2} t^{e^{-t}}dt$

Applying By parts gives

$I=(-te^{-t})_{0}^{ln2}+\displaystyle \int _{0}^{ln2}e^{-t}dt$

$I=-\ln2 \left (\dfrac {1}{2}\right)+(-e^{-t})_0^{ln2}$

$I=\dfrac { -\ln2 }{ 2 } +\left[ \dfrac { -1 }{ 2 } +1 \right] =\dfrac { 1 }{ 2 } -\dfrac { \ln2 }{ 2 }$

$\therefore \ \displaystyle \int_1^2 \dfrac {\ln2}{x^2}dx=\dfrac {1-\ln2}{2}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\int\frac{x+2}{2x^2+6x+5}$
• A. $\frac{1}{4}$ $log(x^2+6x+5)$+ $\frac{1}{2}$$tan^{-1}(2x-3)+C • B. \frac{1}{2} log(2x^2+6x+5)+ \frac{1}{2}$$tan(2x+3)+C$
• C. ${4}$ $log(2x^2+6x+5)$+ $\frac{1}{2}$$tan^{-1}(2x+3)+C • D. \frac{1}{4} log(2x^2+6x+5)+ \frac{1}{2}$$tan^{-1}(2x+3)+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle\int\limits_{1}^{e^2}\dfrac{dx}{x}$
• A. $1$
• B. $-1$
• C. $-2$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle \int {\displaystyle \frac { \sec^{ 2 }x-2010 }{ \sin^{ 2010 }x } dx=\displaystyle \frac { P(x) }{ (\sin x)^{ 2010 } } +c}$ , where $c$ is arbitrary constant then value of $P\left( \displaystyle \frac { \pi }{ 3 } \right)$
• A. $0$
• B. $\displaystyle \frac { 1 }{ \sqrt { 3 } }$
• C. $\displaystyle \frac { 3\sqrt { 3 } }{ 2 }$
• D. $\sqrt { 3 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the rational function   $\cfrac {1}{x(x^4-1)}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\displaystyle \int_{0}^{\pi /2}\frac{dx}{\sin x}$equals
• A. $\displaystyle \frac{1}{2}$
• B. $1$
• C. $3/2$
• D. $0$