# Integration using logs

#### Awsom Guy

Hello I am having trouble with this question.

INtegrate:
f(2x+5)/(x+4) dx

My attempt:
1/2x+5f 1/x+4 dx
2x+5 Ln (x+4) + c

#### galactus

MHF Hall of Honor
$$\displaystyle \int\frac{2x+5}{x+4}dx=2x-3ln(x+4)+C$$

You're close. But there should be a minus where the plus is, and a 3 instead of a 5 in front of the ln.

Awsom Guy

#### Awsom Guy

But how do you get the -3. That is the part I cannot do.
By the way thanks for the help and nice avatar thing.

#### galactus

MHF Hall of Honor
If we expand, we get

$$\displaystyle \frac{2x+5}{x+4}=2-\frac{3}{x+4}$$

Now, integrate that and you can see why.

Awsom Guy

Alternatively,

$$\displaystyle \int{\frac{2x+5}{x+4}}dx=\int{\frac{x+4+x+1}{x+4}}dx=\int{\left(1+\frac{x+1}{x+4}\right)}dx$$

$$\displaystyle u=x+4$$

$$\displaystyle x+1=u-3$$

$$\displaystyle \int{\left(1+\frac{u-3}{u}\right)}du=\int{\left(1+1-3u^{-1}\right)}du=\int{\left(2-3u^{-1}\right)}du$$

$$\displaystyle =2u-3ln|u|+C=2(x+4)-3ln|x+4|+C$$

Awsom Guy