To do integration by substitution using Leibniz notation, we think of the derivative function as a fraction of infinitesimally small quantities du and dx. We change variables by manipulating these infinitesimal quantities.
The general strategy is pretty much the same as before:
Find . 

For the integral, (a) identify u and du and (b) integrate by substitution. 
For the integral, (a) identify u and du and (b) integrate by substitution. 
For the integral, (a) identify u and du and (b) integrate by substitution. 
Find . 
Find . 
Example. Find without using the "multiplying by 1" trick. 
For the integral, (a) identify u and du and (b) integrate by substitution.
For the integral, (a) identify u and du and (b) integrate by substitution.
For the integral, (a) identify u and du and (b) integrate by substitution.
For the integral, (a) identify u and du and (b) integrate by substitution.
For the integral, (a) identify u and du and (b) integrate by substitution.
Integrate by substitution.
Integrate by substitution.
Integrate by substitution.
Integrate by substitution.
Integrate by substitution.
Integrate.
Integrate.
Integrate.
Integrate.
Integrate.