2. Antiderivatives and The Indefinite Integral

by M. Bourne

Mini-Lecture

We wish to perform the opposite process to differentiation. This is called "antidifferentiation" and later, we will call it "integration".

Example

If we know that

and we need to know the function this derivative came from, then we "undo" the differentiation process. (Think: "What would I have to differentiate to get this result?")

y = x³ is ONE antiderivative of math expression

There are infinitely many other antiderivatives which would also work:

y = x³ + 4
y = x³ + π
y = x³ + 27.3

In general: y = x³ + K, is the indefinite integral. K is called the constant of integration.

In electrical engineering, we prefer to write "+K", since C (as used in most math texts) is used for capacitance.

We write: and say:

"The integral of 3x² with respect to x equals x³ + K."

Here is a template you can play with to get the idea.

LIVEMath

The sign is an elongated "S", standing for "sum". Later we will see that the integral is the sum of the areas of infinitely thin rectangles.

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Note: Sometimes we write: F(x) to mean the integral of f(x). So we have:

Exercise: Find

Answer

Note:In general, we can use:

1. (k,K are constants.)

The integral of a constant is that constant times x.

Example:

2. math expression (n ≠ -1)

For the integral of a power of x: add 1 to the power and divide by the new number.

Example:

DON'T FORGET THE "+ K" (or "+ C"). THIS CONSTANT OF INTEGRATION IS VITAL IN LATER APPLICATIONS.

Let's see 2 examples in Flash:

1. How to do basic integration:

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2. Another example:

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Example 1:

Answer

Example 2: math expression

Answer

Example 3:

Answer

Example 4: A particular curve has math expression .

We are told that the curve passes through the point (2, 5). Find the equation of the curve.

Answer

We can an animation of the resulting family of curves in this LiveMath document:

LIVEMath

Example 5:

This is different to the other exercises above! It is a function of a function situation, so we have to do the reverse of the chain rule, which we met in the section on differentiation.

The new rule we require is...

Power Formula for Integration

(n ≠ -1)

Mini-Lecture

See the
mini-lecture on substitution.

This requires a substitution step, where u(x) is some function of x.

Now back to the problem to see how to apply this formula:

Answer

Let's see how it works in Flash:

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Example 6:

Answer

Example 7: Find math expression using a substitution.

Answer

Example 8: Given , find the function y = f(x) which passes through the point (0,2).

Answer

Note: You will see "+K" and "+C" in this work. Most textbooks use + C.

Always use +K if you are answering electrical problems.

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