Question

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

Answer 1

$\int \frac{x^2+x+5}{3x+2}dx$

$\Rightarrow \int \frac{x^2}{3x+2}dx+\int \frac{x}{3x+2}dx+\int \frac{5}{3x+2}dx$

$\Rightarrow \int (\frac{x}{3}-\frac{2}{9}+\frac{4}{9(3x+2)})dx$

$I_1=\int \frac{x}{3}dx+\int \frac{-2}{9}dx+\int \frac{4}{9(3x+2)}dx$

$\Rightarrow \frac{x^2}{6}-\frac{2}{9}x+\frac{4}{9}\int \frac{1}{3x+2}dx$

$\frac{4}{9.3}ln|3x+2|$

$I_2:\int \frac{x}{3x+2}dx$

$\int \frac{1}{9}\left(\frac{U-2}{U}\right)dv$

$\frac{1}{9}U-\frac{2}{9}ln|u|\Rightarrow \frac{1}{9}\left(\right(3x+2)-2ln/3x+2)$

$I_3\Rightarrow \int \frac{5}{3x+2}dx$

$=\frac{5}{3}ln|3x+2|$

$I=\frac{x^2}{6}-\frac{2x}{9}+\frac{4}{27}ln|3x+2|+\frac{1}{9}(3x+2-2n(3x+2)+\frac{5}{3}ln|3x+2|$

$\Rightarrow \frac{x^2}{6}+\frac{1}{9}x+\frac{43}{27}ln|3x+2|+\frac{2}{9}$