Can you show me how to solve first order differential equations using the integrating factor method?

To use the integrating factor method your first order DE must be of the form dy/dx + f(x)y =g(x), where f(x) and g(x) are any functions that depend only on x. lets say f(x)=3x^2 and g(x)=2, (If I feel the tutee would like a greater understanding I would leave f(x) and g(x) arbitrary). Now we define our integrating factor to be e^(integral of x^2) = e^(x^3). Now we multiply our DE by this integrating factor and notice that by using the product rule backwards we get d(e^(x^3)y)/dx =2e^(x^3). (explain this step in more detail by actually computing the left hand side and showing it is equal to what we had beforehand). Now we can use standard integration and rearranging methods to find and equation of y in terms of x. I would now go through more examples with the tutee and progress on to observing them when they try and answer a problem without my help.