Mathematics

# $\displaystyle \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{r=1}^{2n}\frac{r}{\sqrt{n^{2}+r^{2}}}$ equals

$\displaystyle -1+\sqrt{5}$

##### SOLUTION
$\displaystyle \lim _{ n\rightarrow \infty } \dfrac { 1 }{ n } \sum _{ r=1 }^{ 2n } \dfrac { r }{ \sqrt { n^{ 2 }+r^{ 2 } } } =\lim _{ n\rightarrow \infty } \dfrac { 1 }{ n } \sum _{ r=1 }^{ 2n } \dfrac { \dfrac { r }{ n } }{ \sqrt { 1+\dfrac { r^{ 2 } }{ { n }^{ 2 } } } }$

$\displaystyle =\int _{ 0 }^{ 2 }{ \dfrac { x }{ \sqrt { 1+{ x }^{ 2 } } } dx } ={ \left[ \sqrt { 1+{ x }^{ 2 } } \right] }_{ 0 }^{ 2 }=\sqrt { 5 } -1$

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

#### Realted Questions

Q1 Single Correct Medium
$\int _{ 0 }^{ \pi /3 }{ \dfrac { \tan { \theta } }{ \sqrt { 2k\sec { \theta } } } } d\theta =1-\dfrac { 1 }{ \sqrt { 2 } }$. Then $k$ is-
• A. $\dfrac{1}{2}$
• B. $2$
• C. $\dfrac{1}{3}$
• D. $3$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac {ln \left ( x \right )} {x\sqrt{1 + ln \left ( x \right )}} dx$ equlas
• A. $\displaystyle \frac{4}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) - c$
• B. $\displaystyle \frac{1}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) + c$
• C. $\displaystyle 2 \sqrt{1 + ln \left | x \right |} (3ln \left | x \right | - 2) + c$
• D. $\displaystyle \frac{2}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) + c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove:
$\int _ { 0 } ^ { \pi } \dfrac { x \sin x } { 1 + \cos ^ { 2 } x } d x = \dfrac { \pi ^ { 2 } } { 4 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of $\int { { e }^{ { \tan ^{ -1 } { x } } }} \dfrac { \left( 1+x{ +x }^{ 2 } \right) }{ 1{ +x }^{ 2 } } dx$ is ?
• A. $\tan ^{ -1 }{ x } +C$
• B. $e^{\tan^{-2}{x}}+2x+C$
• C. $e^{\tan^{-1}{x}}+C$
• D. ${ xe }^{ \tan ^{ -1 }{ x+C } }$

Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$