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 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."

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.

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 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.

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:

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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