Question

Find the integrals (primitives):
$\rm \displaystyle\int \frac{x^{2}+x+5}{3x+2} \ dx$

Answer 1

Integration by substitution:

$\int \frac{x^{2}+x+5}{3x+2} \ dx\\ \\ let \: \: 3x + 2 = t \\ \\ 3dx = dt \\ \\ dx = \frac{dt}{3} \\ \\ 3x = t - 2 \\ \\ x = \frac{t - 2}{3}$
apply substitution

$\int \frac{( \frac{t - 2}{3}) ^{2}+ \frac{t - 2}{3} +5}{t} \frac{dt}{3} \\ \\ = \int\frac{( \frac{ {t}^{2} + 4 - 4t}{9}) + \frac{t - 2}{3} +5}{t} \frac{dt}{3} \\ \\ = \int \: \frac{{t}^{2} + 4 - 4t+ 3t - 6 +45}{9t} \frac{dt}{3} \\ \\ \frac{1}{27} \int \: \frac{{t}^{2} - t+43}{t}dt \\ \\ \frac{1}{27} \int \: t \: dt - \frac{1}{27} \int \: 1 \: dt - \frac{43}{27} \int \: \frac{1}{t} dt\\ \\ = \frac{ {t}^{2} }{54} - \frac{t}{27} - \frac{43log \: t}{27} + C$
redo substitution

$\int \frac{x^{2}+x+5}{3x+2} \ dx\\\\=\frac{ {(3x + 2)}^{2} }{54} - \frac{(3x + 2)}{27} - \frac{43log \: |3x + 2| }{27} + C$
Hope it helps you.