There are basically 4 techniques for solving an indefinite integral.

1. Directly/by rules.
2. After algebraic manipulation.
3. Using substitution.
4. By parts.

Using limits for improper bounds or internal discontinuities should be mentioned here, but it’s not really in the same category since you don’t use the limit to find an antiderivative.

1. is the method you learn first.
2. includes completing the square, partial fractions, trigonometric integrals where you apply trig identities to whittle down powers or make substitution possible. In a (strong) sense it also includes integration by Taylor series.
3. comes from the chain rule for differentiation, and includes u-substitution and inverse substitution, such as trig substitution.
4. comes from the product rule for differentiation, and is one of the most powerful techniques you learn in calc 2.

When should you apply each technique? There are rules of thumb, though it is also possible to solve many integrals in more than one way.

Completing the square comes in when you have a quadratic polynomial in an awkward spot such as the denominator of a fraction or under a square root. In the latter case it is often followed by trig substitution. In the former case it may lead to a u-substitution where the expression that becomes u has a constant derivative.

Partial fractions are for rational functions, to try to get down to simpler fractions which are amenable to u-substitution or trig substitution.

Trig integrals are easy to identify; they are powers and products of trig functions: sine with cosine, tangent with secant, cotangent with cosecant. They appear frequently after making a trig substitution.

Inverse substitution is generally used when there is an ugly function of x such as a square root, but the rest of the integrand is not simply a constant times the derivative of that function. You set u as a function of x, but solve for x in order to complete the substitution. Square roots of linear functions are a common place for this: if the integrand were , you could let , and then let x = u² – 1 and dx = 2u du. The integrand becomes 2u⁴ – 2u². For trig substitution the traditional variable is theta, and here we exploit similarities to trig identities to collapse binomials into monomials and eliminate problematic square roots or fractions: 1 – sin² x collapses to cos² x, 1 + tan² x to sec² x, and sec² x – 1 to tan² x. [Why introduce sine into the binomial instead of cosine? Because the derivative of your chosen function comes into play in the substitution, and using sine means no negatives are introduced. That is, no reason except it makes for simpler bookkeeping.]

Integration by parts is essential when your integrand is the product of two functions of different “types”: polynomial, trigonometric, exponential, logarithmic.

These techniques can be chained together, of course. Completing the square leading to trig substitution leading to a trig integral is common. Making a substitution to make partial fractions possible is easy to come by (e.g., ).