Tutorial for the Indefinite Integral

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:

Now, a multiple choice question:

x³ dx = ?

3x² + C x⁴

4 + C 3x² dx

x⁴

4 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.

Function

Antiderivative

Formula

xⁿ
(n

xⁿ⁺¹

n+1

+ C

xⁿ dx

xⁿ⁺¹

n+1

+ C (n

Example:

3x^5.4 dx

3x^6.4

6.4

+ C

Function

Antiderivative

Formula

x¹

ln |x| + C

x¹ dx

ln |x| + C

Example:

(5x¹ + 11x³) dx

5 ln |x|

11x²

+ C

Function

Antiderivative

Formula

k
(k constant)

kx + C

k dx

kx + C

Example:

(5x^5.4 + 9) dx

5x^4.4

4.4

+ 9x + C

Function

Antiderivative

Formula

e^x

e^x + C

e^x dx

e^x + C

Example:

(3x^5.4 + 9e^x

4) dx

3x^6.4

6.4

+ 9e^x

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.

(

x²

3x^1.1

= ?

2x²

x³

3x^2.1

2.1

6x + C

2x²

x³

3x^0.1

0.1

+ C

2x²

x³

3x^0.1

0.1

6x + C

2x²

x³

3.3x^0.1

0.1

6x + C

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

5x⁴

x^4.

Then take the antiderivative as above.

In exponent form, the expression 1

6x + x

6 5

4x¹ is ?

x¹

6x¹

20x

6x¹

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

5x²

as either

5x^2/4,

(5x^2)/4,

(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 5e^x should be written as 5e^x, and not 5*e^x.)

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.