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 1
If we know that
`(dy)/(dx)=3x^2`
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 = x^3` is ONE antiderivative of `(dy)/(dx)=3x^2`
There are infinitely many other antiderivatives which would also work, for example:
`y = x^3+4`
`y = x^3+pi`
`y = x^3+27.3`
In general, we say `y = x^3+K` is the indefinite integral of `3x^2`. The number K is called the constant of integration.
Note: Most math text books use `C` for the constant of integration, but for questions involving electrical engineering, we prefer to write "+K", since C is normally used for capacitance and it can get confusing.
Notation for the Indefinite Integral
We write: `int3x^2dx=x^3+K` and say in words:
"The integral of 3x2 with respect to x equals x3 + K."
Here is a template you can play with to get the idea.
The Integral Sign
The `int` 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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Other Notation for Integrals
Note: Sometimes we write a capital letter to signify integration. For example, we write F(x) to mean the integral of f(x). So we have:
`F(x)=intf(x)dx`
Example 2
Find `int(x^2-5)dx`
We now learn some important general rules for integration.
A. Integral of a Constant
`intk\ dx=kx+K`
(`k` and `K` are constants.)
The integral of a constant is that constant times x.
Example 3
Find `int4\ dx`
B. Integral of a Power of x
`intx^ndx=(x^(n+1))/(n+1)+K` (This is true as long as `n ≠ -1`)
For the integral of a power of x: add 1 to the power and divide by the new number.
Example 4
Integrate `intx^5 dx`
The Constant of Integration
Don't forget the "+ K" (or, alternatively, "+ C"). This constant of integration is vital in later applications of the indefinite integral.
The next 2 examples use Flash to show the steps:
Example 5: How to do basic integration
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Example 6
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Example 7
`int7x^6dx`
Example 8
`int(3x^2+sqrtx-5/x^3)dx`
Example 9
`introot(3)(x^2)dx`
Example 10
A particular curve has its derivative given by `(dy)/(dx)=3x^2-2x`.
We are told that the curve passes through the point (2, 5). Find the equation of the curve.
We can see an animation of the resulting family of curves in this LiveMath document:
Example 11
Consider this integration:
`int(2x^4-5)^6x^3dx`
This is different to the other exercises above!
The expression we have to integrate containts `(2x^4-5)^6`, which is a function of a function, and we have that `x^3` at the end. We cannot do this integration using the rules we have learned so far.
In this case, we have to do the reverse of the Chain Rule, which we met in the section on differentiation.
We introduce a new rule for integrating cases like these.
C. Power Formula for Integration
`int u^ndu=u^(n+1)/(n+1)+K`
(This is true if `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.
Integrate `int(2x^4-5)^6x^3dx`.
Let's see how it works in Flash:
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More examples
Example 12
`int(x^3-2)^6(3x^2)dx`
Example 13
Find `intx/(sqrt(x^2+9))dx` using a substitution.
Example 14
Given `y^'=sqrt(2x+1`, 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.
It's a good idea to always use +K if you are answering electrical problems.
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