What's the difference betwen parameterizations and variable substitution for solving integrals? Asumming I have the following integral to solve in the complex plane:
$$\int \frac{dz}{z+1} $$ while $|z|=5$ which means a contour of radius 5 around zero.
Is it possible to solve this integral using:
$$z+1=5e^{it}$$ while $t$ runs from $0$ to $2\pi$ ?
if this move is legit, what excactly am I doing? is it a parameterization or is it a varbile substitution?
 A: That doesn't seem right.
$$z+1=5e^{it}$$
isn't a correct parameterization of the circle with radius 5 and the center as origin. 
A correct parameterization would be: $z=z(t) = 5e^{it}$ which would lead to the integral:
$$\int_0^{2\pi}\frac{1}{5e^{it}+1}\cdot ie^{it} \text{d}t$$
Is it a variable substitution?
No, then you would need to apply the 'transformation of contour integration'-theorem:

If $w$ holomorphic on $\Gamma$ and $f$ continous on $w(\Gamma)$ then
  $$\int\limits_{w(\Gamma)} f(w) \operatorname dw = \int\limits_\Gamma f(w(z)) w'(z) \operatorname dz$$

Which would imply that you need to change the contour itself when transforming $z+1=5e^{it}$
How do I solve it?
There are various ways of solving this. You could use the residue theorem (which seems a bit overkill here).
Another possibility is to use the transformation of contour integrals and transforming $z +1 =w$, where the contour would shift 1 to the left.
Since the new contour still encloses the origin, you probably know the result: $2\pi i$
