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
Every antiderivative of 2x has the form x2 + C, where C is constant. |
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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
Examples
Reading the Formula Here is how we read the first formula above:
The constant of integration, C, reminds us that we can add any constant and get a different antiderivative. Some For You |
Here is a multiple choice quiz:
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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 ≠ -1) |
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Function | Antiderivative | Formula | ||||||||||||||
x-1 | ln |x| + C |
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Function | Antiderivative | Formula | ||||||||||||||
k (k constant) |
kx + C |
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Function | Antiderivative | Formula | ||||||||||||||
ex | ex + C |
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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.
Q | How do we deal with powers of x in the denominator, such as in, say, | 5x4 | ? |
5x4 | as | 5 | x-4. |
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\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 blanks and press "Check."
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