2. Antiderivatives and The Indefinite Integral
by M. Bourne
Mini-Lecture
See the
mini-lecture on differentials
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 
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.
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.
Loading Flash movie...
Note: Sometimes we write: F(x) to mean the integral of f(x). So we have:
Exercise: Find 
Note:In general, we can use:
1. 
(k,K are
constants.)
The integral of a constant is that constant times x.
Example: 
2. 
(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:
Loading Flash movie...
2. Another example:
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Example 1: 
Example 2: 
Example 3: 
Example 4: A particular curve has 
.
We are told that the curve passes through the point (2, 5). Find the equation of the curve.
We can an animation of the resulting family of curves in this LiveMath document:
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:
Let's see how it works in Flash:
Loading Flash movie...
Example 6: 
Example 7: Find 
using a substitution.
Example 8: Given 
, find the function y =
f(x) which passes through the point (0,2).
Note: You will see "+K" and "+C" in this work. Most textbooks use + C.
Always use +K if you are answering electrical problems.
Find your integral using Mathematica!
Enter multiply using *, square root of x using Sqrt[x] and trigonometry like Sin[x]. See the full list of how to enter math.
Click the blue button to find your integral.
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