Mathematics

Evaluate: $\displaystyle \int x\log(1+x) dx$

$\dfrac{(2x^2 -2)\ln(x+1)-x^2 +2x}{4}+c$

SOLUTION
$\int { x } \log { \left( 1+x \right) } dx$

$=\log { \left( 1+x \right) } \int { x } dx-\int { \left[ \dfrac { d }{ dx } \log { \left( 1+x \right) } .\int { x.dx } \right] }$

$=\log { \left( 1+x \right) } \dfrac { { x }^{ 2 } }{ 2 } -\int { \dfrac { 1 }{ 1+x } . } \dfrac { { x }^{ 2 } }{ 2 } dx$

$=\dfrac{x^2}{2}\log \left( x+1\right)-\dfrac{1}{2}\int \dfrac{x^2-1+1}{\left(x+1\right)}dx$

$=\dfrac{x^2}{2}\log \left(x+1\right) -\dfrac{1}{2}\left[ \int { \dfrac { \left( x-1 \right) \left( x+1 \right) }{ \left( x+1 \right) } } dx+\int { \dfrac { 1 }{ \left( x+1 \right) } dx } \right]$

$=\dfrac{x^2}{2}\log \left(x+1\right)-\dfrac{1}{2}\int \left(x-1\right)dx-\dfrac{1}{2}\log \left(x+1\right)$

$=\dfrac{1}{2}\log \left( x+1\right)\left(x^2-1\right)-\dfrac{1}{2}\left[ \dfrac { { x }^{ 2 } }{ 2 } -x \right] +c.$

$=\dfrac{1}{2}\log \left( x+1\right)\left(x^2-1\right)-\dfrac{1}{2}\left[ \dfrac { { x }^{ 2 } }{ 2 } -x \right] +c.$

Hence, the answer is $\dfrac{(2x^2 -2)\ln(x+1)-x^2 +2x}{4}+c$

Its FREE, you're just one step away

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

Realted Questions

Q1 Single Correct Hard
$\displaystyle \int_{1}^{e^{37}}\frac{\pi \sin \left ( \pi \log _{e}x \right )}{x}dx$ is equal to
• A. $\displaystyle -2$
• B. $\displaystyle 2/\pi$
• C. $\displaystyle 2\pi$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int { \frac { { x }^{ 2 } }{ { \left( 2+{ 3x }^{ 2 } \right) }^{ 5/2 } } dx }$ is equal to
• A. $\dfrac { 1 }{ 6 } { \left[ \dfrac { { x }^{ 2 } }{ 2+3{ x }^{ 2 } } \right] }^{ 3/2 }+C$
• B. $\dfrac { 1 }{ 6 } { \left[ \dfrac { { x }^{ 2 } }{ 2+3{ x }^{ 2 } } \right] }^{ 7/2 }+C$
• C. $None\ of\ these$
• D. $\dfrac { 1 }{ 5 } { \left[ \dfrac { { x }^{ 2 } }{ 2+3{ x }^{ 2 } } \right] }^{ 3/2 }+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate $\displaystyle \overset { 7 }{ \underset { 4 }{ \int } } \dfrac{(11 - x)^2}{x^2 + ( 11 - x)^2} dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate $\int {{\dfrac{{e^{{{\tan }^{ - 1}}x}}}{1 + {x^2}}}dx,x \in \left( {0,\infty } \right)}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The value of $\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\dfrac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$ is
• A. $\pi$
• B. $0$
• C. none of these
• D. $\dfrac{\pi}{4}$