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

$$\int \frac {x+4}{ 2x+6 } dx$$

This is a problem from Khan Academy that I was reading about how to solve when I accidentally clicked next and lost the explanation. I was reading something about how there is a clever way to divide the function to make it easier to integrate. Can someone please explain this to me?

No actual solution, please. I want to get it by myself.

Solution 1:

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

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

Solution 2:

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.

Solution 3:

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.

Solution 4:

After seeing $ 2 x+ 6 $ in denominator and then the moment you see $x+4$ in numerator it should be doubled and halved as..

$$\int\frac{x+4}{2x+6}dx$$

$$=\frac12 \int\frac{2 x+8}{(2 x + 6 )}dx$$

$$=\frac{x}{2}+ \int\frac{dx}{2 x + 6}$$

$$=\frac{x}{2}+ \frac12 \int\frac{dx}{ x + 3},$$

it can be continued.