## 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 Select one x^4+C 4x^3 is Select one x^4+C 4x^3 .
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

Here is how we read the first formula above:

 2x dx = x2 + C The antiderivative of 2x, with respect to x, equals x2 + 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+1n+1 + C
 xn dx = xn+1n+1 + C     (n ≠ -1)
 Examples: x5.4 dx = x6.46.4 + C Use the formula with n = 5.4 3x5.4 dx = 3x6.46.4 + 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-22 + C
FunctionAntiderivativeFormula
k
(k constant)
kx + C
 k dx = kx + C
 Example: (5x-5.4 + 9) dx = - 5x-4.44.4 + 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 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 - x22 + 3x-1.1 - 6) dx
=

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

 65x4 as 65 x-4.
Then take the antiderivative as above; for instance, the antiderivative of this expression would be
\int \frac{6}{5}x^{-4} dx = \frac{6}{5}\frac{x^{-3}}{-3} + C = -\frac{6x^{-3}}{15} + C

In exponential form, the expression is:

Fill in the blank and press "Check."

 ( 7ex + 16x + x6 - 54x-2 ) 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 Section 6.1 of Applied Calculus or Section 13.1 of Finite Mathematics and Applied Calculus

Last Updated: July, 2008