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
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