Mathematics

# $\displaystyle \int \dfrac{2x+3}{3x+2}dx$ is equal to

##### SOLUTION
Given the integral,
$\int { \dfrac { 2x+3 }{ 3x+2 } } dx$
Let,
$u=3x+2\\ \Rightarrow \dfrac { du }{ dx } =3\\ \Rightarrow dx=\dfrac { 1 }{ 3 } du$
Substituting these values in the given integral we get,
$\int { \dfrac { 2x+3 }{ 3x+2 } } dx\\ =\dfrac { 1 }{ 9 } \int { \dfrac { 2u+5 }{ u } } du$
For,
$\int { \dfrac { 2u+5 }{ u } } du\\ =\int { (\dfrac { 5 }{ u } +2) } du\\ =5\int { \dfrac { 1 }{ u } } du+2\int { (1) } du\\ =5\ln { (u) } +2u\\ \therefore \dfrac { 1 }{ 9 } \int { \dfrac { 2u+5 }{ u } } du\\ =\dfrac { 5\ln { (u) } }{ 9 } +\dfrac { 2u }{ 9 } \\ =\dfrac { 5\ln { (3x+2) } }{ 9 } +\dfrac { 2(3x+2) }{ 9 }$
Hence, $\int { \dfrac { 2x+3 }{ 3x+2 } } dx=\dfrac { 5\ln { (3x+2) } }{ 9 } +\dfrac { 2(3x+2) }{ 9 } +C.$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Let $y = y(x), y(1)=1\ and\ y(e) ={e^2}$ . Consider
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