## 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. )

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## 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:

 2x dx = x2 + C The antiderivative of 2x, with respect to x, equals x2 + 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:

 6 dx =

Now, a multiple choice question:

 x3 dx = ?
 3x2 + C x44 + C 3x2 dx
 x44 dx

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+1n+1 + C
 xn dx = xn+1n+1 + C     (n 1)
 Example: 3x5.4 dx = 3x6.46.4 + C
FunctionAntiderivativeFormula
x1 ln |x| + C
 x1 dx = ln |x| + C
 Example: (5x1 + 11x3) dx = 5 ln |x| 11x22 + C
FunctionAntiderivativeFormula
k
(k constant)
kx + C
 k dx = kx + C
 Example: (5x5.4 + 9) dx = 5x4.44.4 + 9x + C
FunctionAntiderivativeFormula
ex ex + C
 ex dx = ex + C
 Example: (3x5.4 + 9ex 4) dx = 3x6.46.4 + 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.

 ( 4x x22 + 3x1.1 6) dx = ?

 2x2 x36 3x2.12.1 6x + C
 2x2 x36 3x0.10.1 + C
 2x2 x36 3x0.10.1 6x + C
 2x2 x33 3.3x0.10.1 6x + C

Question
 How do we deal with powers of x in the denominator, such as in, say, 65x4 ?

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

 65x4 as 65 x4.
Then take the antiderivative as above.

 In exponent form, the expression 16x + x6 54x1 is ?

 x16 + x6 5x4
 6x1 + x6 20x
 6x1 + x6 5x4
 x6 + x6 + 5x4

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

 5x24 as either 5x^2/4, (5x^2)/4, or (5/4)x^2, but not 5/4x^2.

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.)

 ( 7ex + 16x + x6 54x2 ) dx
=

You now have several options

• Try some of the questions in the true/false quiz (warning: it covers the whole of Chapter 6) by pressing the button on the sidebar.
• Try some of the on-line review exercises (press the "review" button on the sidebar. Again, these questions cover the whole chapter, but Questions 1(a) and 2(a) are relevant.)
• Try some of the exercises on pp. 441-442 of Calculus Applied to the Real World, or pp. 939-940 ofFinite Mathematics and Calculus Applied to the Real World.

Last Updated: April, 1997