Mathematics

# The value of the integral $\displaystyle \int _{ { e }^{ -1 } }^{ { e }^{ 2 } }{ \left| \frac { \log _{ e }{ x } }{ x } \right| dx }$ is

$\dfrac {3}{2}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Using integral $\int _{ 0 }^{ \pi /2 }{ \ln { \left( \sin { x } \right) } } dx=\int _{ 0 }^{ \pi /2 }{ \ln { \left( \sec { x } \right) } } dx=-\cfrac { \pi }{ 2 } \ln { 2 }$
Evaluate $\int _{ -\pi /4 }^{ \pi /4 }{ \ln { \left( \cfrac { \sin { x } +\cos { x } }{ \cos { x } -\sin { x } } \right) } dx= }$
• A. $\pi \ln{2}$
• B. $\cfrac{\pi \ln{2}}{2}$
• C. $-\pi \ln{2}$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
The value of integral $\displaystyle\int^{\pi /2}_{\pi_{/4}}{\cos x}dx$ is?

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate: $\displaystyle\int_{0}^{\dfrac{\pi}{2}}\sqrt{sin\,\phi} \, cos^5\,\phi\,d\,\phi$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $\int { \cfrac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+4 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }-5 \right) } } dx=\int { \left\{ 1+\cfrac { f(x) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }-5 \right) } \right\} } dx$
$x+A\tan ^{ -1 }{ \left( \cfrac { x }{ A' } \right) } +B\log { \left( \cfrac { x-l }{ x+m } \right) } +K\quad$ then which of the following is correct
• A. $A=\cfrac { 1 }{ 4\sqrt { 3 } } ,B=\cfrac { 27 }{ 8\sqrt { 5 } } ,K\in R$
• B. $f(x)=7{ x }^{ 2 }+19,A'=\sqrt { 3 } ,K\in R$
• C. $l=m=\sqrt { 5 } ,L=1,K\in R$
• D. All of these

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.