Integration Techniques & Improper Integrals
Core tricks of single-variable integration — substitution, parts, partial fractions — plus improper integrals and their applications.
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Questions Covered in This Set
11 cards to master
State the substitution rule and what it reverses.
∫f(g(x))g'(x)dx = F(g(x)) + C, where F' = f. It is the chain rule read backwards; set u = g(x), du = g'(x)dx.
Why change the limits in a definite-integral substitution?
Converting limits to u-values (e.g. x=0→u=0, x=1→u=1 for u=x²) lets you finish in u without back-substituting — fewer places to slip.
The three standard trig substitutions
√(a²−x²): x = a sinθ; √(a²+x²): x = a tanθ; √(x²−a²): x = a secθ. Beware √(x²)=|x| when inverting.
Integration by parts formula and its origin
∫u dv = uv − ∫v du, from integrating the product rule (uv)' = u'v + uv'. You trade one integral for a hopefully easier one.
What does LIATE stand for and what is it for?
Log, Inverse trig, Algebraic, Trig, Exponential — choose u from the earliest category when doing integration by parts.
Compute ∫ln x dx
u = ln x, dv = dx ⇒ v = x, du = dx/x: x ln x − ∫1 dx = x ln x − x + C.
The 'loop trick': ∫eˣ sin x dx
Parts twice gives I = eˣ sin x − eˣ cos x − I, so I = ½eˣ(sin x − cos x) + C — solve algebraically for the integral.
Partial fraction decomposition: what forms appear?
For deg P < deg Q, P/Q splits into A/(x−r)^k and (Bx+C)/(x²+bx+c)^k; each integrates to a log, a power, or an arctangent. If deg P ≥ deg Q, divide first.
Heaviside cover-up shortcut
After clearing denominators, plug in the roots. E.g. 3x+1 = A(x+1)+B(x−2): x=2 gives A=7/3, x=−1 gives B=2/3.
Two ways an integral becomes improper, and what convergence means
Infinite interval or unbounded integrand. Define by a limit (∫₁^∞ f = lim_{b→∞}∫₁^b f); if the limit exists finitely it converges, otherwise it diverges.
Liouville's theorem punchline
Every rational function has an elementary antiderivative, but ∫e^(−x²)dx provably has none — knowing unwinnable battles is part of maturity.
