Guides · 8 min read
How to Know Which Integration Technique to Use
Differentiation is a procedure: follow the rules and you always get an answer. Integration is a search. There is no single algorithm that solves every integral, which is why students often stare at a problem with no idea where to begin. The good news is that the handful of standard techniques each have clear fingerprints, and learning to spot them turns that blank stare into a short checklist.
This guide is organized around recognition rather than mechanics. For each method it describes the shape of integral that calls for it, so that your first move on any problem is diagnosis, not guesswork.
First, Try to Recognize It Directly
Before reaching for any technique, check whether the integral is already a known antiderivative or a trivial rewrite away from one. The power rule for integrals, the basic trig integrals, and the exponential and logarithm forms cover a surprising share of textbook problems. Splitting a sum into separate integrals and pulling out constants often reduces a scary-looking expression to a list of standard pieces.
The lesson here is to not overcomplicate. Many students launch into integration by parts on an integral that only needed the power rule after distributing. Simplify the integrand algebraically first, then see what remains. Only when no direct antiderivative is visible do you move to the techniques below.
u-Substitution: Look for a Function and Its Derivative
u-substitution is the reverse of the chain rule, and it is the technique you try first among the real methods. The fingerprint is a composite function paired with something close to the derivative of its inner part. If you can spot an inner function whose derivative also appears in the integrand, up to a constant, substitution will collapse the whole thing into a simple integral in u.
Concretely, watch for an expression raised to a power, or under a root, or inside a trig or exponential function, while its derivative sits elsewhere as a factor. Set u equal to the inner function, replace dx with du over the derivative, and the integral usually simplifies dramatically. Substitution is the most frequently used technique in the entire subject, so make it your default first attempt.
u-substitution undoes the chain rule when the inner derivative is present.
Integration by Parts: A Product of Two Unlike Functions
When the integrand is a product of two functions of genuinely different types, and substitution does not apply because neither factor is the derivative of the other, integration by parts is usually the answer. The classic candidates are a polynomial times an exponential, a polynomial times a trig function, or a logarithm or inverse trig function standing on its own.
The method trades your integral for a hopefully easier one, using the rule that the integral of u dv equals u times v minus the integral of v du. The art is choosing which factor to call u. The common mnemonic LIATE, standing for Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential, ranks which factor should be u, earlier letters first. Pick u so that its derivative gets simpler, because you want the leftover integral to be more manageable than the one you started with.
Integration by parts, best for products of unlike function types.
Trigonometric Substitution: A Square Root of a Sum or Difference of Squares
Trig substitution is the specialist you call when you see a square root containing a sum or difference of squares, such as the root of (a squared minus x squared) or (x squared plus a squared). These forms resist ordinary substitution because there is no inner derivative to exploit. Instead you replace x with a trig function chosen so that a Pythagorean identity collapses the root into a single term.
The three canonical patterns each match a trig identity: use sine for a squared minus x squared, tangent for x squared plus a squared, and secant for x squared minus a squared. After the substitution the integral becomes a trig integral, which you solve with identities and then convert back using a reference triangle. It is the most involved technique, so reserve it for integrals whose square roots leave you no simpler option.
Partial Fractions: A Rational Function You Cannot Integrate Whole
When the integrand is a ratio of polynomials that you cannot integrate directly, partial fractions is the tool. The idea is to reverse the process of adding fractions: break one complicated rational function into a sum of simpler fractions, each with a linear or quadratic denominator, that you can integrate one at a time. The result is typically a combination of logarithms and arctangents.
Two setup steps matter. First, if the numerator's degree is greater than or equal to the denominator's, do polynomial long division first and integrate the polynomial part separately. Second, factor the denominator completely, because the form of your partial-fraction decomposition depends on whether the factors are distinct, repeated, or irreducible quadratics. Once the algebra is set up, the integration itself is easy.
A Quick Decision Checklist
Put the techniques together into a fast triage you can run on any integral. The order reflects both frequency and effort, cheapest and most common first.
- Can you simplify or recognize a standard antiderivative? Do that first.
- Is there an inner function with its derivative present? Use u-substitution.
- Is it a product of two unlike functions? Use integration by parts.
- Is it a rational function of polynomials? Use partial fractions (after long division if needed).
- Does it contain a square root of a sum or difference of squares? Use trigonometric substitution.
- Still stuck? Look for an algebraic rewrite or a trig identity that changes the form entirely.
Frequently asked questions
Which integration technique should I try first?
After checking for a direct antiderivative, always try u-substitution first. It is the most common technique and the fastest when it works, since it simply reverses the chain rule.
How do I choose u in integration by parts?
Use the LIATE order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Whichever type appears earliest in that list becomes u, so that its derivative simplifies.
When do I use trig substitution instead of u-substitution?
Use trig substitution when the integrand contains a square root of a sum or difference of squares and there is no inner derivative available for a plain substitution.
Do I always need partial fractions for rational functions?
Not always. If the denominator is a simple power or the numerator is its derivative, substitution may work. Reserve partial fractions for ratios you cannot integrate whole, and do long division first if the top degree is too high.