Mathematics

Integrate:$$\displaystyle\int { \dfrac { { x }^{ 2 }+x+5 }{ 3x+2 }  } dx$$


SOLUTION
$$ \displaystyle \int \frac{x^2+x+5}{3x+2}dx $$
$$ \displaystyle \Rightarrow \int \frac{x^{2}}{3x+2}dx+\int \frac{x}{3x+2}dx+ \int \frac{5}{3x+2}dx$$
$$\displaystyle  \Rightarrow \int (\frac{x}{3}-\frac{2}{9}+\frac{4}{9(3x+2)})dx$$
$$ \displaystyle I_1 = \int \frac{x}{3}dx+\int \frac{-2}{9}dx+\int \frac{4}{9(3x+2)}dx$$
$$\displaystyle \Rightarrow \frac{x^2}{6}-\frac{2}{9}x+\frac{4}{9}\int \frac{1}{3x+2}dx$$
$$\displaystyle  \frac{4}{9.3}ln\left | 3x+2 \right |$$
$$ \displaystyle I_2 : \int \dfrac{x}{3x+2}dx $$
$$ \displaystyle \int \dfrac{1}{9}(\dfrac{U-2}{U})dv $$
$$ \displaystyle \frac{1}{9}U-\dfrac{2}{9}ln|u|\Rightarrow \dfrac{1}{9}((3x+2)-2ln/3x+2)$$
$$ \displaystyle I_3 \Rightarrow \int \dfrac{5}{3x+2}dx $$
$$ \displaystyle  = \dfrac{5}{3}ln |3x+2|$$
$$ \displaystyle I = \dfrac{x^{2}}{6}-\dfrac{2x}{9}+\dfrac{4}{27}ln|3x+2|+\dfrac{1}{9}(3x+2-2n(3x+2)+\dfrac{5}{3} ln |3x+2|$$
$$ \displaystyle \Rightarrow \frac{x^{2}}{6}+\dfrac{1}{9}x+\dfrac{43}{27}ln|3x+2|+\dfrac{2}{9}$$
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Subjective Medium Published on 17th 09, 2020
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