# 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: \[
\text{for }\bar x_i=\frac{x_{i-1}+x_i}{2}=a+\frac{(2i-1)(b-a)}{2n},\text{ we get }\int_a^bf(x)\,\mathrm dx\approx\sum_{i=1}^nf(\bar x_i)\Delta x.
\]