Question

integrate with respect to x:
5x-2/1+2x+3x^2

plzzzzzzzzzzzzzz fast friends!
super urgent!!!

Answer 1

$f(x)=\frac{5x-2}{1+2 x+3x^2}\\1+2x+3x^2=3[(x+\frac{1}{3})^2+(\frac{\sqrt{2}}{3})^2]=\frac{2}{9}[y^2+1],\\where\ y=\frac{x+\frac{1}{3}}{\frac{\sqrt2}{3}}=\frac{3}{\sqrt2}x+\frac{1}{\sqrt2}\\\\ dy= \frac{3}{\sqrt2}dx\\\\x=\frac{\sqrt2 y-1}{3}\\5x-2=5\sqrt2 y/3 - 11/3\\\\ \int\limits^{}_{} {f(x)} \, dx = \int\limits^{}_{} {\frac{15}{\sqrt2}\frac{y}{y^2+1}} \, dy - \int\limits^{}_{} {\frac{3}{22(y^2+1)}} \, dy$

Substitute for y.

answer=
$\frac{15}{2\sqrt2} Ln (1+3x^2+2x) - K -\frac{3}{22}Tan^{-1}\frac{3x+1}{\sqrt2},\\K=integration\ constant$