Section 2.10  A Summary of Integration Techniques
Now that you have encountered most of the techniques of integration, it's time to summarize and identify when to use what. You should first note that the advice I give here is only a set of rough guidelines.


Starting and Thinking Two Steps Ahead
"Starting" is the process of listing out different methods that you deem possible. It's a good idea to write out the integration techniques in two lists: the possible and the not likely. The "not likely" list will contain techniques that you immediately know will not work. For example, if you were asked to evaluate the integral \[\int{\frac{x^2+2}{x^3  8} dx},\] it should be pretty obvious that trigonometric integration techniques are unnecessary. Thus, we would throw "trig int" into the "not likely" list; however, substitution or partial fraction decomposition may seem like possibilities, so we would throw those into the "possible" list.
"Thinking two steps ahead" is the process of understanding the consequences of your method of choice and seeing whether or not that method will be fruitful. The more integrals you evaluate, the better equipped you will be to start thinking two steps ahead. In the integral listed above, some may say that splitting the integral into \[\int{\frac{x^2}{x^3  8} dx} + 2\int{\frac{1}{x^3  8} dx}\] will allow a useful substitution of \(u = x^3  8\) in the first integral. While this may be true because \(du = 3x^2 dx\), that second integral still requires partial fraction decomposition. Therefore, you might as well just use PFD from the getgo.
Be aware that you often must use several of the techniques listed below in a single integral. Moreover, don't be afraid to abandon a tactic that does not seem to be working out. Start over with a different technique and only go back to your original tactic if all else fails.
Simplify the Integrand
You have successfully gone through the entire algebra sequence along with a full course in trigonometry  it's time to use those skills! Simplify the integrand using trigonometric identities or algebraic manipulations, if possible. You may be surprised to learn how easy it would have been to evaluate that integral had you just done some simple manipulations at the beginning.
Simple integration
"Starting" is the process of listing out different methods that you deem possible. It's a good idea to write out the integration techniques in two lists: the possible and the not likely. The "not likely" list will contain techniques that you immediately know will not work. For example, if you were asked to evaluate the integral \[\int{\frac{x^2+2}{x^3  8} dx},\] it should be pretty obvious that trigonometric integration techniques are unnecessary. Thus, we would throw "trig int" into the "not likely" list; however, substitution or partial fraction decomposition may seem like possibilities, so we would throw those into the "possible" list.
"Thinking two steps ahead" is the process of understanding the consequences of your method of choice and seeing whether or not that method will be fruitful. The more integrals you evaluate, the better equipped you will be to start thinking two steps ahead. In the integral listed above, some may say that splitting the integral into \[\int{\frac{x^2}{x^3  8} dx} + 2\int{\frac{1}{x^3  8} dx}\] will allow a useful substitution of \(u = x^3  8\) in the first integral. While this may be true because \(du = 3x^2 dx\), that second integral still requires partial fraction decomposition. Therefore, you might as well just use PFD from the getgo.
Be aware that you often must use several of the techniques listed below in a single integral. Moreover, don't be afraid to abandon a tactic that does not seem to be working out. Start over with a different technique and only go back to your original tactic if all else fails.
Simplify the Integrand
You have successfully gone through the entire algebra sequence along with a full course in trigonometry  it's time to use those skills! Simplify the integrand using trigonometric identities or algebraic manipulations, if possible. You may be surprised to learn how easy it would have been to evaluate that integral had you just done some simple manipulations at the beginning.
Simple integration
 Identifying: This tactic is very useful if the integrand is really close to the derivative of a function you know.
 Some Warnings: Keep an eye out for simple algebraic manipulations or trigonometric identities in the integrand as performing them often leads to this form!


Substitution
Partial fraction decomposition (PFD)
By parts
 Identifying: Substitution (a.k.a. "\(u\)substitution") is useful in many cases; however, the two most popular are
1) if there are two functions in the integrand and one is pretty much the derivative of the other, or
2) there is a composition of functions in the integrand.  Some Warnings: Always remember to convert back to the original variable in the end (for indefinite integrals) or change the limits of integration (for definite integrals) right when you do the substitution.
Partial fraction decomposition (PFD)
 Identifying: Of all the integration techniques, partial fraction decomposition tends to be the easier to identify. If the integrand is a rational expression and a substitution will not work, then this is the method for you!
 Some Warnings: This technique can lead to some algebraically heavy computations  especially when the denominators contain irreducible quadratic factors. Be prepared to beef up your matrix algebra skills or learn to use the matrix feature in a graphing calculator.
 Additional Advice: It is often the case that this method leads to integrals of the form \[\int{\frac{1}{ax + b} dx} = \frac{1}{a} \ln{ax + b} + C\] and \[\int{\frac{1}{x^2 + a^2} dx} = \frac{1}{a} \tan^{1}{\left(\frac{x}{a}\right)} + C.\] These are great integrals to remember.
By parts
 Identifying: Typically, this is a great method if the integrand is a product of different types of functions. You can try to use LIATE (Logarithmic functions, Inverse trigonometric functions, Algebraic functions, Trigonometric functions, Exponential functions) to help choose the value for \(u\) (the variable you are going to take the derivative of), but this sometimes fails. Another, somewhat better, tactic is to choose \(u\) to be the function whose derivative is "less complicated" (e.g. lower power) or of a different form than the original (e.g. \(\frac{d}{dx}\left(\ln{x}\right) = \frac{1}{x}\)).
 Some Warnings: If the integrand contains functions whose derivatives do not "decay" (e.g. polynomials and/or radicals), then you may have to use this method a couple times, backtoback. See the section on Integration By Parts to see what I mean.
 Additional Advice: Many integrals can be evaluated using this method that do not match the "product of different functions" rule. I suggest to always consider this as a tactic to try on a nonobvious case when all else fails.


Products of Trigonometric Functions (just fancy substitutions)
Trigonometric Substitution
Radical Substitution (just fancy substitution)
 Identifying: These, like partial fraction decomposition problems, tend to be somewhat obvious. They have the form \[\int{\sin^m{(x)} \cos^n{(x)} dx}\] or \[\int{\sec^m{(x)} \tan^n{(x)} dx}.\] Just remember to try a substitution that forces the other trigonometric function to end with even powers after the change from \(dx\) to \(du\). After that, just use trigonometric identities (specifically, the Pythagorean Identities).
 Some Warnings: If the powers are already even on those sines and cosines, then you must resort to using the halfangle identities from trigonometry to convert everything into cosines. This can get messy if the even powers are large. Additionally, these types of integrals are common targets for extra \(u\)substitutions and by parts methods – be wary! Finally, if the arguments on the trigonometric functions are not the same (e.g., \(\int{\sin{(3x)} \cos{(5x)} dx}\)), then you must resort to using the somewhat obscure ProducttoSum formulas from trigonometry.
Trigonometric Substitution
 Identifying: I would love to say these are obvious, but many students actually overuse this technique. If the integrand contains a sum or difference of squares (under a radical or not) and a simple substitution will not work, then this is a great tactic. Make the appropriate substitution for \(x\) so that you get one of the following Pythagorean identities from trigonometry: \(1  \sin^2{(\theta)} = \cos^2{(\theta)}\), \(1 + \tan^2{(\theta)} = \sec^2{(\theta)}\), or \(\sec^2{(\theta)}  1 = \tan^2{(\theta)}\).
 Some Warnings: Sometimes you must complete the square for a polynomial expression in the integrand to get it into a form like a sum or difference of squares.
Radical Substitution (just fancy substitution)
 Identifying: This technique is very powerful. If the integrand contains a mixture of polynomials and a radical of the form \(\sqrt[n]{g(x)}\), allowing \(u=\sqrt[n]{g(x)}\) is an awesome tactic.
 Additional Advice: With this method, you almost always have to solve for \(x\) in \(u=\sqrt[n]{g(x)}\) to get the other expressions in your integrand in terms of \(u\). Moreover, I highly suggest not directly finding \(du\) from \(u=\sqrt[n]{g(x)}\), but instead using implicit differentiation on \(u^n = g(x)\) to get a "cleaner" form of the differential.

