Section 6.1: The Indefinite Integral

(Based on Section 6.1 inCalculus Applied to the Real World, or Section 13.1 inFinite Mathematics and Calculus Applied to the Real World. )

If you got here directly from the outside world and see no frames, press here to bring up the frames that will allow you to properly navigate this tutorial and site.

Notes for Microsoft Explorer users:

1. You may notice vertical alignment glitches and table formatting errors here and there -- Explorer often gets confused with complicated tables containing large numbers of images, subscripts, and/or superscripts, and many of our mathematical formulas use pictures in tables within tables.
2. Some versions of Explorer don't know how to direct Javascript commands to the proper window in a frame. If some responses to your answers take up the whole page, just press the "back" button on your browser to go back to the tutorial.

If you find this annoying, let Microsoft know, or switch to Netscape to see what this page is supposed to look like.

Note To understand this section, you should be familiar with derivatives. Press the "index" button on the sidebar to select one of the on-line tutorials on derivatives.

For best viewing, adjust the window width to at least the length of the line below.

Indefinite Integrals, or Antiderivatives

Antiderivative

An antiderivative of a function f(x) is just a function whose derivative is f(x).

Example

Since the derivative of x2+4 is 2x, an antiderivative of 2x is x2+4.
Since the derivative of x2+30 is also 2x, another antiderivative of 2x is x2+30.
Similarly, another antiderivative of 2x is x249.
Similarly, another antiderivative of 2x is x2 + C, where C is any constant (positive, negative, or zero) In fact:

Every antiderivative of 2x has the form x2 + C, where C is constant
Notation

We write

    2x dx = x2+C

Here is how we read the formula:

2xdx=x2 + C
The antiderivative of 2x, with respect to x, equalsx2 + C

Fill in the blanks and press "Check." (Write x3 as x^3, andx2 as x^2, and don't forget to include the "dx" and the "+C" in the proper places. Include as many spaces as you want; they will be ignored.)

Since the derivative of x^3 is 3x^2,

Now do one yourself:

Now, a multiple choice question:

The correct answer to the last question suggests a formula for finding the antiderivative of any power of x. The following table includes this formula, as well as other information.

FunctionAntiderivativeFormula
xn
(n 1)
xn+1

n+1
+ C
xn dx=
xn+1

n+1
+ C     (n 1)
Example: 3x5.4 dx=
3x6.4

6.4
+ C
FunctionAntiderivativeFormula
x1 ln |x| + C
x1 dx=ln |x| + C
Example: (5x1 + 11x3) dx=5 ln |x|
11x2

2
+ C
FunctionAntiderivativeFormula
k
(k constant)
kx + C
k dx=kx + C
Example: (5x5.4 + 9) dx=
5x4.4

4.4
+ 9x + C
FunctionAntiderivativeFormula
ex ex + C
ex dx=ex + C
Example: (3x5.4 + 9ex 4) dx=
3x6.4

6.4
+ 9ex 4x + C

If you would like a hard copy of the above table, press here to obtain a new page which you can then print out.

Question
How do we deal with powers of x in the denominator, such as in, say,
6

5x4
?

Answer
First convert them into exponent form; that is, rewrite the expression with all powers of x in the numerator. For example, rewrite

Then take the antiderivative as above.

In exponent form, the expression
1

6x
+
x

6
5

4x1
is ?

Fill in the blank and press "Check." Use standard calculator formatting; for example, write

and write ln |x| just like it is written here. Spaces will be ignored.
(Do not use "*" at all; for example 5ex should be written as 5e^x, and not 5*e^x.)

You now have several options

Last Updated: April, 1997
Copyright © 1997 Stefan Waner and Steven R. Costenoble