Techniques of integration
The idea
Differentiation is mechanical, but integration is pattern recognition: most integrands must be transformed before any basic antiderivative applies. The core toolkit is substitution (the chain rule run backwards), integration by parts (the product rule run backwards), partial fractions for rational functions, and trigonometric substitution for roots of quadratics. Fluency here is what makes later courses — differential equations, probability, physics — feel routine instead of impossible.
Integration by parts comes from rearranging (uv)' = u'v + uv': the integral of u dv equals uv minus the integral of v du. The strategy is to trade a hard integral for an easier one, so choose u as the factor that simplifies when differentiated (polynomials, logarithms) and dv as the factor you can integrate cleanly (exponentials, sines, cosines). A common misconception is that any split works; choose badly and the new integral is worse than the old one — picking u = e^(2x) in the example below would do exactly that.
Build the habit of pattern-scanning before computing: a function next to its own derivative suggests substitution, a product of unrelated types suggests parts, a ratio of polynomials suggests partial fractions. The first thirty seconds of looking usually saves ten minutes of algebra.
Worked example
Evaluate the integral of x e^(2x) from x = 0 to x = 1.
- Scan the structure: x and e^(2x) are unrelated types and substitution finds no inner-derivative pair, so integration by parts is the tool.
- Choose u = x because differentiating it simplifies to du = dx, and dv = e^(2x) dx because it integrates cleanly to v = e^(2x)/2.
- Apply the parts formula: the integral becomes x e^(2x)/2 minus the integral of e^(2x)/2 dx — note the new integral has no polynomial factor, confirming the trade was a good one.
- Finish the remaining integral: e^(2x)/2 antidifferentiates to e^(2x)/4, so an antiderivative is F(x) = x e^(2x)/2 − e^(2x)/4.
- Evaluate from 0 to 1: F(1) = e²/2 − e²/4 = e²/4, and F(0) = 0 − 1/4 = −1/4, so the integral is e²/4 + 1/4 = (e² + 1)/4 ≈ 2.097.
- Sanity-check: the integrand rises from 0 to e² ≈ 7.39 over a unit interval and is convex, so an answer around 2.1 — well below the midpoint estimate of 3.7 — is reasonable.
Answer. The integral of x e^(2x) from 0 to 1 is (e² + 1)/4 ≈ 2.097.
Check your understanding
- Why is integration by parts described as the product rule run in reverse, and where does the minus sign come from?
- What goes wrong, concretely, when you pick u = e^(2x) and dv = x dx in this example?
- How do you recognize within seconds that an integrand calls for substitution rather than parts?
- What features of an integrand would push you toward partial fractions, and what must you check about the degrees first?
Build the foundations first
Techniques of integration builds on these concepts. If any feel shaky, start there.