#### Passage

The average value of a function f(x) over the interval, [a,b] is the number $\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$
The square root $\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$ is called the root mean square of f on [a, b]. The average value of $\displaystyle \mu$ is attained id f is continuous on [a, b].
Mathematics

# The average value of f(x)=$\displaystyle \frac{\cos ^{2}x}{\sin ^{2}x+4\cos ^{2}x}on\left [ 0,\pi /2 \right ]$ is

1/6

##### SOLUTION
$\cfrac { 1 }{ \pi /2-0 } \int _{ 0 }^{ \pi /2 }{ \cfrac { \cos ^{ 2 }{ x } }{ \sin ^{ 2 }{ x } +4\cos ^{ 2 }{ x } } dx }$
Substituting $t=\tan { x }$
$=\cfrac { 2 }{ \pi } \int _{ 0 }^{ \infty }{ \cfrac { dt }{ \left( { t }^{ 2 }+4 \right) \left( { t }^{ 2 }+1 \right) } } \\ =\cfrac { 2 }{ \pi } \cfrac { 1 }{ 3 } \int _{ 0 }^{ \infty }{ \left( \cfrac { 1 }{ { t }^{ 2 }+1 } -\cfrac { 1 }{ { t }^{ 2 }+4 } \right) dt } \\ =\cfrac { 1 }{ 6 }$

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Single Correct Hard Published on 17th 09, 2020
Mathematics

# The average value of the pressure varying from 2 to 10 atm if the pressure p and the volume v are related by $\displaystyle pv^{3/2}=160$ is

$\displaystyle \frac{40}{3\sqrt{20}\left ( 3\sqrt{10}+3\sqrt{2} \right )}$

##### SOLUTION
As p varies from 2 to 10 atm, v transverses the interval $\left[ 4\sqrt [ 3 ]{ 4 } ,4\sqrt [ 3 ]{ 100 } \right]$
Average value =$=\cfrac { 1 }{ 4\left( \sqrt [ 3 ]{ 100 } -\sqrt [ 3 ]{ 4 } \right) } \int _{ 4\sqrt [ 3 ]{ 4 } }^{ 4\sqrt [ 3 ]{ 100 } }{ 160{ v }^{ -3/2 }dv } \\ =\cfrac { 40 }{ \sqrt [ 3 ]{ 20 } \left( \sqrt [ 3 ]{ 10 } +\sqrt [ 3 ]{ 2 } \right) }$

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Single Correct Medium Published on 17th 09, 2020
Mathematics

# The average ordinate of y= sin x over the interval $\displaystyle \left [ 0,\pi \right ]$ is

$\displaystyle 2/\pi$

##### SOLUTION
$\mu =\cfrac { 1 }{ x } \int _{ 0 }^{ \pi }{ \sin { x } dx } =-\cfrac { 1 }{ x } { \left[ \cos { x } \right] }_{ 0 }^{ \pi }=\cfrac { 2 }{ \pi }$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 One Word Medium
If  $\int \dfrac{1+\sin x}{\sin x (1+\cos x) }dx=\dfrac{1}{2}\ln \left|\tan \left(\dfrac{x}{2}\right)\right|+\dfrac{1}{m}\tan ^2\left(\dfrac{x}{2}\right)+\tan \left(\dfrac{x}{2}\right)+C$.Find $m$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\int_{0}^{1} e^{x^{2}}(x-\alpha) d x=0,$ then
• A. $\alpha<0$
• B. $0<\alpha<1$
• C. $\alpha=0$
• D. $1<\alpha<2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int {\sqrt {\dfrac{{x - 5}}{{x - 7}}dx} }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of integral $\displaystyle \int \frac{\left(\sqrt{1+x^{2}}+x \right)^{n}}{\sqrt{1+x^{2}}}dx$, is
• A. $\displaystyle \frac{1}{n}\left ( \sqrt{1+x^{2}}+x \right )^{n-1}+c$
• B. $\displaystyle \frac{1}{n-1}\left ( \sqrt{1+x^{2}}+x \right )^{n-1}+c$
• C. $\displaystyle \frac{1}{n-1}\left ( \sqrt{1+x^{2}}+x \right )^{n}+c$
• D. $\displaystyle \frac{1}{n}\left ( \sqrt{1+x^{2}}+x \right )^{n}+c$

$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$