How can I integrate $\int{1\over 2x+2}$

Question

$$\int{1\over 2x+2}$$

Method 1

$$\int{1\over 2x+2} = \frac 12\int{1\over x+1} = \frac 12 ln(x+1) + c $$

Method 2

$$\int{1\over 2x+2} = \frac 12\int{2\over 2x+2} = \frac 12 ln(2x+2) + c $$

Wolfram Alpha suggests the first method yields the correct answer, but why do the two methods produce different answers?

Answer 1

You can do this $$\int{1\over 2x+2}dx = \frac 12\int{dx\over x+1} = \frac{1}{2}\int{\frac{du}{u}} = \frac{1}{2}\ln{u} + c$$ where $u = x + 1$, $du = dx$. You know that $$ \int{\frac{du}{u}} = \ln{u} + c $$