Cognitive Process for Integration

The following are the concepts, rules, and technicalities you must consider as you approach each new integral.

To hone your integration skills sufficiently for upper-level academic work and professional work, you must practice these skills until they are indelibly inscribed in your brain.

There is no other way to achieve the requisite degree of skill with integration but to practice incessantly. However, if you consistently practice for just an hour or two every day, the process will seem more like a series of short jogs than a grueling marathon.

Pre-Integration Considerations

Before you even pick up your pen(cil) to work on an integral, think about these important considerations:

1. Is the integral proper? Are there any singularities within the interval of integration? Is the interval of integration finite?
2. Is the lower limit of integration greater than the upper limit? If so, then you must reverse them and remember to throw a negative sign out front. $$\int_2^1 f(x) dx = -\int_1^2 f(x) dx$$
3. Is the integrand an odd function which "straddles" the Y-axis? If so, then the area between the curve and the X-axis on one side of the Y-axis cancels the area between the curve and the X-axis on the other side. $$\int_{-a}^{a} f(x) dx = 0$$
4. Is the integrand an even function which "straddles" the Y-axis? If so, is it possible that you only need to integrate half of the interval defined by the limits of integration?
   • If it is the case that the integrand is an even function, and you only need to integrate over half the interval, be sure to multiply the integral by 2. $$\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$$
5. Which integration technique will work best with it? What type of function is the integrand?

Overarching Strategy

According to the author of the (awesome) CalculusFTW app, there are essentially $3$ integration techniques:

1. Change the way the integrand looks This includes using algebra, partial fraction decomposition, and trigonometric identities.

2. Substitution This includes the substitution rule (i.e. U-Substitution), rationalizing substitutions, and inverse substitutions like trigonometric substitutions.

3. Integration by Parts