Mathematics

# $n\overset{Lt}{\rightarrow}\infty \displaystyle \{\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{6n}\}=$

log 6

##### SOLUTION

$n\overset{Lt}{\rightarrow}\infty\sum_{r=1}^{5n}\dfrac{1}{n+r}=n\overset{Lt}{\rightarrow}\infty\sum_{r=1}^{5n}\dfrac{1}{n}\left [ \dfrac{1}{1+\left ( \dfrac{r}{n} \right )}\right ]$
It is in the form
$n\overset{Lt}{\rightarrow}\infty\sum_{r=1}^{n}\dfrac{1}{n}f\left ( \dfrac{r}{n} \right )=\int_{0}^{1}f(x)dx$
So, $n\overset{Lt}{\rightarrow}\infty\sum_{r=1}^{5n}\dfrac{1}{n}f\left ( \dfrac{r}{n} \right )=\int_{0}^{5}\dfrac{1}{1+x}dx$
$=log (1+x)\int_{0}^{5}$
$=log (6)-log (1)$
$=log 6$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Hard
If $\displaystyle I = \int \frac {dx}{(2 \sin x + \sec x)^4}$, then I equals
• A. $\displaystyle \frac {1}{5 \tan^5 x} + \frac {1}{3 \tan^6 x} - \frac {I}{(2 \sin x + \sec x)^3} + C$
• B. $\displaystyle \frac {-1}{3(2 \sin x + \sec)^3} + \tan^{-1} (3\sqrt {\tan x}) + C$
• C. $\displaystyle \frac {-1}{3(2 \sin x + \sec x)^3} - \tan^{-1} (3\sqrt {\tan x}) + C$
• D. $\displaystyle -\frac {1}{5 \tan^5 x} + \frac {1}{3 \tan^6 x} - \frac {2}{7 \tan^7 x} + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac { 1 - x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) \sqrt { 1 + x ^ { 4 } } } d x$ is equal to
• A. $\sqrt { 2 } \sin ^ { - 1 } \left\{ \frac { \sqrt { 2 } x } { x ^ { 2 } + 1 } \right\} + c$
• B. $\frac { 1 } { 2 } \sin ^ { - 1 } \left\{ \frac { \sqrt { 2 } x } { x ^ { 2 } + 1 } \right\} + c$
• C. $\frac { 1 } { \sqrt { 2 } } \sin ^ { - 1 } \left\{ \frac { x ^ { 2 } + 1 } { \sqrt { 2 } x } \right\} + c$
• D. $\frac { 1 } { \sqrt { 2 } } \sin ^ { - 1 } \left\{ \frac { \sqrt { 2 } x } { x ^ { 2 } + 1 } \right\} + c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int { \frac { dx }{ x\left( { x }^{ 7 }+1 \right) } } \\$ is equal to:
• A. $\displaystyle \log { \left( \frac { { x }^{ 7 } }{ { x }^{ 7 }+1 } \right) }$
• B. $\displaystyle \log { \left( \frac { { x }^{ 7 }+1 }{ { x }^{ 7 } } \right) } +c$
• C. $\displaystyle \frac { 1 }{ 7 } \log { \left( \frac { { x }^{ 7 }+1 }{ { x }^{ 7 } } \right) } +c$
• D. $\displaystyle \frac { 1 }{ 7 } \log { \left( \frac { { x }^{ 7 } }{ { x }^{ 7 }+1 } \right) } +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Solve :
$\displaystyle \int \frac{1}{x+x\log x}dx.$
• A. $\displaystyle \frac{1}{2}\log \left ( 1+\log x \right ).$
• B. $\displaystyle \log \left ( \log x \right ).$
• C. $\displaystyle \log \left ( 1-\log x \right ).$
• D. $\displaystyle \log \left ( 1+\log x \right ).$

$\int \frac{2x^{2}}{3x^{4}2x} dx$