Integrals¶
Recall (doc/postdoc). Not a first course.
Job: antiderivative vs definite integral; Riemann first, Lebesgue for limits under integral sign.
Core¶
FTC: if \(F'=f\) continuous on \([a,b]\), then \(\int_a^b f=F(b)-F(a)\). \(F(x)=\int_a^x f\) is \(C^1\) with \(F'=f\).
| Technique | Identity |
|---|---|
| Substitution | \(\int f(g(x))g'(x)\,dx=\int f(u)\,du\) |
| Parts | \(\int u\,dv=uv-\int v\,du\) |
| Partial fractions | rational \(p/q\) |
| Trig / Weierstrass | \(t=\tan(x/2)\) |
Improper: \(\lim_{b\to\infty}\int_a^b\) or singular endpoints; absolute vs conditional convergence.
Comparison: \(0\le f\le g\) and \(\int g<\infty\Rightarrow\int f<\infty\).
Used later for¶
- Measure theory: Lebesgue, DCT/MCT
- Multivariable: Fubini, Jacobian
- Complex analysis: contours
- Probability: \(\mathbb{E}[X]=\int x\,dF\)
- ODE/PDE: variation of parameters, Duhamel
Check¶
Why can one interchange \(\frac{d}{dx}\int_a^b f(x,t)\,dt=\int_a^b\partial_x f\) only under extra hypotheses?
Need domination / uniform convergence of difference quotients (or \(C^1\) + compact interval). Pointwise \(\partial_x f\) existence alone does not justify differentiation under the integral sign.