Strategy Summary for Integration

# Strategy Summary for Integration

We have so far looked at 6 varying techniques of integration, namely:

In each section we assumed to apply a certain technique according to the section. There will often be times when you don't know which method to apply at first, so we will summarize a strategy for tackling on any integral.

 Step 1: Simplify the integrand as much as possible. Sometimes a daunting integral can be made to be extremely simple with some algebraic simplification. For example, consider integrating the function $f(x) = \frac{x + 2}{x + 1}$. Integrating this example isn't particularly hard, however, notice that $f(x) = \frac{(x + 1) + 1}{(x+ 1)} = 1 + \frac{1}{x + 1}$. Integrating $f(x)$ now is even simpler!
 Step 2: Look for an easy u-substitution. U-substitution is probably one of the easier techniques to apply, so it is definitely a first choice if it is easy to apply. For example, integrating the function $f(x) = \frac{x + 1}{x^2 + 1}$ can be tackled on more easily by making the substitution $u = x^2 + 1$ so that $du = 2x \: dx$.
 Step 3: If the function is a product of two functions and if one of the functions when differentiated gets "simpler", then attempt the technique of integration by parts.
 Step 4: Recognize any indicators to utilize other integration tricks. If the function is a product of cosine, sine, etc…, then applying advanced trigonometric integration techniques will be useful. If the function is a product where one term differentiated multiple times goes to zero, then tabular integration might be useful. If a function contains a radical, then trigonometric substitution might be recommended. Lastly, if the function is a quotient of two polynomials, then integration by partial fractions might work. Sometimes it may be necessary to use multiple methods to evaluate an integral.
 Step 5: If none of the earlier steps lead you anywhere, then attempt a method of integration that you assume is most likely to work. It is important to note however, that some functions are note integrable, so don't get down if you can't integrate what isn't even integrable!