Mathematics

# The value of $\displaystyle\int { \cfrac { 1 }{ x+x\log { x } } } dx$

$\log { (1+\log { x } ) }$

##### SOLUTION
$I=\displaystyle\int{\dfrac{dx}{x+x\log{x}}}$

$=\displaystyle\int{\dfrac{dx}{x\left(1+\log{x}\right)}}$

Let $t=1+\log{x}\Rightarrow\,dt=\dfrac{1}{x}dx$

$=\displaystyle\int{\dfrac{dt}{t}}$

$=\log{\left(t\right)}+c$     .......where $c$ is constant of integration

$=\log{\left(1+\log{x}\right)}+c$    .........where $t=1+\log{x}$

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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

Solve $\int {\dfrac{{{x^5}}}{{\sqrt {1 + {x^2}} }}} \,dx$

• A. $\dfrac{1}{{15}}\sqrt {1 + {x^2}} \left( {3{x^4} + 4{x^2} + 8} \right) + C$
• B. $\sqrt {1 + {x^2}} \left( {3{x^4} + 4{x^2} + 8} \right) + C$
• C. None of these
• D. $\dfrac{1}{{15}}\sqrt {1 + {x^2}} \left( {3{x^4} - 4{x^2} + 8} \right) + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let f (x) be function satisfying $f'(x)=f(x)$ with $f(0)=1$ and g be the function satisfying f(x)+g(x)=$\displaystyle x^{2}.$ The value of the integral $\displaystyle \int_{0}^{1}f\left ( x \right )g\left ( x \right )dx$ is
• A. $\displaystyle e-\frac{1}{2}e^{2}-\frac{5}{2}$
• B. $\displaystyle e-e^{2}-3$
• C. $\displaystyle \frac{1}{2}\left ( e-3 \right )$
• D. $\displaystyle e-\frac{1}{2}e^{2}-\frac{3}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\int _{ 0 }^{ 3 }{ \left( 2{ x }^{ 2 }+3x+5 \right) dx }$ as limit of a sum.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the function    $\sqrt {1-4x^2}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int { \cfrac { \cos { x } -1 }{ \sin { x } +1 } } { e }^{ x }dx$ is equal to:
• A. $c-\cfrac { { e }^{ x }\sin { x } }{ 1+\sin { x } }$
• B. $c-\cfrac { { e }^{ x }}{ 1+\sin { x } }$
• C. $c-\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } }$
• D. $\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } } +c$