By the end of this lesson, you should be able to:

2. Find the antiderivative or indefinite integral of a function.

3. Find the definite integral of a function with and without using a calculator.

Integration

Integration is a calculus operation that lets one find the integral of a function. Recall from differential calculus that derivative of a function was the instantaneous rate of change of that function, illustrated by the slope of a line tangent to the curve of the function at a given point. Integral calculus deals with the integral of a function, which can be illustrated as the area under the curve of a function within a given interval. Actually, this is just the "signed" area, or the area between the curve and the horizontal axis.

Integration and differentiation are inverse. Velocity is the derivative of distance traveled with respect to time, and distance traveled is the integral of velocity with respect to time. The basic relationship here is between the slope of a curve and the area between that curve and the horizontal axis.

Indefinite Integral or Antiderivative

Thus, we can compute an indefinite integral or antiderivative of a function by working backward through the process of differentiation. Consider the following:

The symbol on the left means "the integral of" and it ends with the dx on the right, which means "with respect to x."

So, what function has 6x² as its derivative? Let's see, we reduced the power by one, so we must have started with x³, but what was its coefficient?  If we started with  x³, then we multiplied the original coefficient by 3, therefore the original coefficient must have been 2. But was the original function

f (x) = 2 x ³,
f (x) = 2 x ³ + 12, or
f (x) = 2 x ³ - 20 ?

Any of these would have produced the same derivative. Let's just say that the integral of 6 x ² is 2 x ³ plus a constant (C). Sometimes the antiderivative is written as a capital F. F is the antiderivative of f:

F(x) = 2 x ³ + C

Note that the antiderivative or indefinite integral turns out to be a function, not a value or number.

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What is the antiderivative of 5 x ³ + 2 x ?

Reveal the Answer => (5/4) x ⁴ + x ² + C

Definite Integral

Integration is commonly used to find the area between the curve of a function and the horizontal axis. Please note that not all functions are continuous. Tan(x), for example, has vertical asymptotes that result in discontinuity. Thus, while tan(x) is integrable for certain ranges, it is not integrable over other ranges.

For functions that are integrable over a range (a to b), we can compute the definite integral by subtracting the antiderivative of a from the antiderivative of b.

Let's say we wish to find out the area under the curve of the function y = 0.1x ² + 0.4x, and we are just interested in the area between x = 2 and x = 5.

We use the integration sign but with the subscript 2 and the superscript 5 to indicate the range.

As above, we find F(x), which turns out to be:
(1/30) x ³ + .2 x ² + C
(if you'll pardon the mixture of fractions and decimals)

Now, let's substitute 5 for x, and then subtract what we get when we substitute 2 for x

((1/30) 5 ³ + .2 * 5 ² ) - ((1/30) 2 ³ + .2 * 2 ² ) = 8.1

This can also be found and illustrated using some graphing calculators. Notice how the area under the curve is shaded for the range specified, and the answer value of the definite integral appears at the bottom of the TI-89 screen shown here. Be sure to see if your calculator allows you to find and graph integrals.

Note that the definite integral turns out to be a value or number, not a function.

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An object is traveling in a straight line with a velocity (in feet per second) of

v = 3t ³ + 5t ²
at time t.

How far has the object traveled between t = 1 and t = 7 seconds?

Highlight the:

1. Indefinite Integral => (3/4)t ⁴+ (5/3)t ³ + C
2. Answer =>2370 feet