Indefinite integrals and the Net change theorem

Indefinite integrals

  • To express the connection between antiderivatives and definite integrals supplied by the FTC, an integral without boundaries denotes the general antiderivative of the integrand
    • Definition. Let \(f(x)\) be a function. Then the indefinite integral \[ \int f(x)\,\mathrm dx=F(x)+C \] is the general antiderivative, if it exists.
    • For example, we have \[ \int x^2-3\sin x+x^{-1}\,\mathrm dx=\frac{x^3}{3}+3\cos x+\ln|x|. \]
    • Exercises. 5.4: 6, 16, 18, 34, 45, 46, 49, 50

The Net change theorem and the Midpoint rule

  • The Net change theorem is a reformulation of FTC part 2, which is useful for applications.
    • Theorem. If \(F\) is a differentiable function on an open interval containing \([a,b]\), then we have \[ \int_a^bF'(x)\,\mathrm dx=F(b)-F(a). \]
    • Exercises. 5.4: 60, 63, 64
  • The Midpoint rule is saying that for a general function \(f(x)\) (that if, if we don't have additional information about it), to get a good approximation, you can use midpoints as sample points: \[ \text{for }\bar x_i=\frac{x_{i-1}+x_i}{2}=a+\frac{(2i-1)(b-a)}{2n},\text{ we get }\int_a^bf(x)\,\mathrm dx\approx\sum_{i=1}^nf(\bar x_i)\Delta x. \]
    • Exercise. 5.4: 66