Mathematics

# $\int {\dfrac{{\log \left( {x + 1} \right) - \log x}}{{x\left( {x + 1} \right)}}} \,\,dx$ equals

##### SOLUTION
$\int { \dfrac { log\left( x+1 \right) -logx }{ x\left( x+1 \right) } dx }$
Put $log\left( x+1 \right) -logx=4\\ \left( \dfrac { 1 }{ x+1 } -\dfrac { 1 }{ x } \right) dx=dt\\ \dfrac { -1 }{ x\left( x+1 \right) } dx-dt\\ \dfrac { dx }{ x\left( x+1 \right) } =-dt$
$\int { -tdt } \\ =\dfrac { -{ t }^{ 2 } }{ 2 } +c\\ =\dfrac { -\left[ log\left( x+1 \right) -logx \right] ^{ 2 } }{ 2 } +c$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral as limit of sum:
$\displaystyle \int_{0}^{5}(x+1)dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle\int^{\pi}_0\sqrt{|\cos x|-|\cos^3x|}dx$ is?
• A. $1$
• B. $3$
• C. $4$
• D. $\dfrac{4}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate:$\displaystyle \int _{ 0 }^{ \pi /2 }{ \cos ^{ 2 }{ x } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { \cfrac { 3{ x }^{ 5 } }{ 1+{ x }^{ 12 } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
If $n\rightarrow \infty$ then the limit of series in $n$ can be evaluated by following the rule : $\displaystyle \lim_{n\rightarrow \infty}\sum_{r=an+b}^{cn+d}\frac{1}{n}f\left ( \frac{r}{n} \right )=\int_{a}^{c}f(x)dx,$
where in $LHS$, $\dfrac{r}{n}$ is replaced by $x$,
$\dfrac{1}{n}$ by $dx$
and the lower and upper limits are $\lim_{n\rightarrow \infty }\dfrac{an+b}{n}\, and \, \lim_{n\rightarrow \infty }\dfrac{cn+d}{n}$ respectively.