# 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