Simple Rules for Differentiation



Indefinite Integrals

Objectives:

Students will be able to

• Calculate an indefinite integral.

• Calculate the constant of integration.

So far we have found derivatives of functions. Each function has only one derivative. Multiple functions can have the same derivative.

Sometimes we want to be able to go the other direction. Undoing differentiation is called antidifferentiation. The process of antidifferentiation is often finding the indefinite integral. The integral that we find is indefinite because we are unable to determine which of the multiple functions produced the derivative we started with. These multiple functions are often referred to as a class of function and they only differ by the addition of a constant term.

The way we write this is

[pic] where [pic]

In this form, the symbol [pic]is the integral sign; f(x) is the integrand; x is the variable of integration; and C is the constant of integration.

Several standard and important integrals come from some of the simple rules for differentiation.

Important Integrals

• [pic]

• [pic]

• [pic]

• [pic]

Some general rules about integrals arise from general rules about derivatives.

General Rules of Integration

• [pic] where a is a constant

• [pic]

So far we have talked about finding the class of functions that when differentiated will have the function we are integrating. What happens if we want to know exactly what the original function is? Generally in order to do this, we need further information about the function. Usually this information is a point that the function goes through. This will allow us to determine the actual value of the constant of integration.

You should also know that just like it was possible to find higher order derivatives by differentiating multiple times, it is also possible to integrate multiple times.

Example 1:

Evaluate the indefinite integral [pic]

Example 2:

Evaluate the indefinite integral [pic]

Example 3:

Evaluate the indefinite integral [pic]

Example 4:

Evaluate the indefinite integral [pic]

Example 5:

Evaluate the indefinite integral [pic]

Example 6:

Evaluate the indefinite integral [pic]

Example 7:

Evaluate the indefinite integral [pic]

Example 8:

Under certain conditions, the number of diseased cells N(t) at time t increases at a rate [pic], where A is the rate of increase at time 0 (in cells per day) and k is a constant.

a. Suppose A = 60, and at 4 days, the cells are growing at a rate of 300 per day. Find a formula for the number of cells after t days, given that 400 cells are present at t = 0.

b. Use your answer from part a to find the number of cells present after 11 days.

Example 9:

Suppose [pic], [pic], and [pic]. Find s(t).

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