So this was on my test today and I couldnt figure it out.
int[ (x+2) / (x+5) ]
obviously part of the answer was ln|x+5|, but i couldnt figure out what to do with the (x+2)
thanks in advance!
$\displaystyle \int\frac{x+2}{x+5}\,dx$
$\displaystyle =\int\frac{x+2\color{red}+3-3}{x+5}\,dx$
$\displaystyle =\int\frac{x+5-3}{x+5}\,dx$
$\displaystyle =\int\left(\frac{x+5}{x+5}-\frac3{x+5}\right)dx$
$\displaystyle =\int\left(1-\frac3{x+5}\right)dx$
$\displaystyle =\int dx-3\int\frac1{x+5}\,dx$
$\displaystyle =x-3\ln|x+5|+C$