How do I evaluate $\int \frac {x+4}{ 2x+6 } dx $?

HINT: Notice $$\int\frac{x+4}{2x+6}dx$$ $$=\int\frac{x+4}{2(x+3)}dx$$ $$=\frac{1}{2}\int\frac{x+4}{x+3}dx$$ $$=\frac{1}{2}\int\frac{(x+3)+1}{x+3}dx$$

$$=\frac{1}{2}\int\left(1+\frac{1}{x+3}\right)dx$$

The change of variable $t=x+3$ simplifies the denominator and turns the integral in

$$\frac12\int\frac{t+1}tdt,$$ which should now be obvious.

The way you have to think about it is like this. $$\int \frac{x+4}{2x+6}\ dx$$ then factor the denominator to get $2(x+3)$ then pull one half out the integral to get $$ \frac{1}{2}\int \frac{x+4}{x+3}\ dx$$. here is the cleverness rewrite $x+4$ as $x+3+1$ then simplify to get $$ \frac{1}{2}\int \left(1+ \frac{1}{x+3}\right) dx$$ Hope that helps.