6.1 The Indefinite Integral

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

Antiderivative

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

Examples

  • 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 x2-49.
  • 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.

Q Since the derivative of x4+C is 4x3,

    an antiderivative of is .    
Indefinite Integral

We call the set of all antiderivatives of a function the indefinite integral of the function. We write the indefinite integral of the function f as

    \int f(x) dx
and we read it as "the indefinite integral of f(x) with respect to x" Thus,\int f(x) dx is a collection of functions; it is not a single function, nor a number. The function f that is being integrated is called the integrand, and the variable x is called the variable of integration.

Examples

    \int 2x dx = x^2 + C     The indefinite integral of 2x with respect to x is x2 + C
    \int 4x^3 dx = x^4 + C     The indefinite integral of 4x3 with respect to x is x4 + C

Reading the Formula

Here is how we read the first formula above:

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

The constant of integration, C, reminds us that we can add any constant and get a different antiderivative.

Some For You

Q Since the derivative of 4x3 is 12x2,

    =

Q Another one:

    6 dx =


 

Here is a multiple choice quiz:

= ?

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)
Examples: x5.4 dx=
x6.4

6.4
+ C   Use the formula with n = 5.4
3x5.4 dx=
3x6.4

6.4
+ C   The multiple 3 "goes along for the ride"
FunctionAntiderivativeFormula
x-1 ln |x| + C
x-1 dx=ln |x| + C
Example: (5x-1 + 11x-3) dx=5 ln |x|-
11x-2

2
+ C
FunctionAntiderivativeFormula
k
(k constant)
kx + C
k dx=kx + C
Example: (5x-5.4 + 9) dx=-
5x-4.4

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

For this quiz, you need to enter an algebraic expression using proper graphing calculator format as above (spaces are ignored). Press the button to see examples including expressions involving logarithms and exponentials.

(x2 - 3x-1 + 4)dx
=
(2x-1.1 + 0.5x0.5 + 2ex)dx
=
(4x-
x2

2
+3x-1.1 - 6)dx
=

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

5x4
?
  A First convert them into exponential form; that is, rewrite the expression with each summand in the form Ax^n, where A and n are constants. For example, rewrite

Then take the antiderivative as above; for instance, the antiderivative of this expression would be

In exponential form, the expression is:


 

Fill in the blank and press "Check."

You now have several options

Last Updated: July, 2008
Copyright © 2008 Stefan Waner

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