Alright so I'm a little confused with basic integration.
$\displaystyle \int\frac{dx}{x}=ln|x|+c$
But what if theres a constant infront of dx.
For example
$\displaystyle \int\frac{7dx}{x}=\int\frac{1}{x}*7dx=?$
What changes then?
As 7 is a constant, it is not affected by the integration, so you can do this:
$\displaystyle
\int\frac{7dx}{x}$
$\displaystyle = 7\int\frac{dx}{x}$
$\displaystyle = 7ln|x| + c$
$\displaystyle = ln |x^7| + c
$
or if you like to think of it this way:
$\displaystyle 7\int\frac{7dx}{x}(\frac{1}{7})$
$\displaystyle
\int\frac{7dx}{4x+2}
$
You multiply by the derivative, but don't forget to divide it.
For this one, I'll take the 7 out again and use the substitution u = 4x + 2. du = 4 dx.
You need a 4 on top, so multiply by $\displaystyle \frac{4}{4}$.
$\displaystyle
7\int\frac{4dx}{4x+2}(\frac{1}{4})
$
Since 1/4 is a constant, I took it out like I did with the 7. (Note: only constants that are multiplied or divided can be taken out, addition or subtraction of constants are integrated.)
$\displaystyle \frac{7}{4} \int\frac{du}{u}$ (since du = 4dx; u = 4x+2)
=$\displaystyle \frac{7}{4} ln|4x+2| +c$