1. Integrals
2. Seed Dispersal Problem
3. Differential Equations

This section contains a number of examples that require the use of integration by parts. These examples include some standard integrations, a seed dispersal problem, and an example using a differential equation.

Example 1: Integrate the following example:

Solution: When an integral has a polynomial multiplying an exponential, sine, or cosine function, then this integral can be solved using integration by parts with the polynomial piece (in this case x) being selected for the u(x) and the other part (in this case sin(3x)) being selected for the v'(x). By letting u(x) = x, then u'(x) = 1. With v'(x) = sin(3x), we integrate this to obtain

From the integration by parts formula, it follows that

Example 2: Suppose that a plant is found to disperse seeds in an exponentially decaying manner. In the range from 3-4 meters, the experimenter discovers 20 seeds

Solution: W

Example 3: Integrate the following example:

Solution: W