Antidifferentiation

Antidifferentiation is the reverse process of differentiation.

If the derivative of [latex]x^2[/latex] is [latex]2x[/latex] , then we can say that an antiderivative of [latex]2x[/latex] is [latex]x^2[/latex]. That is to ask, what function must we differentiate to get [latex]2x[/latex]? The answer (or at least an answer) is [latex]x^2[/latex].

Just as we can state that if [latex]f(x) = x^2[/latex]  ,  [latex]f'(x)=2x[/latex] , (that is the derivative of [latex]x^2[/latex] is [latex]2x[/latex]) , we could also say that if [latex]f(x)=2x[/latex]  then [latex]F(x)=x^2[/latex] , (an antiderivative of [latex]2x[/latex] is [latex]x^2[/latex]).

Of course if you differentiate [latex]f(x)=x^2[/latex] , [latex]f(x)=x^2+1[/latex] , [latex]f(x)=x^2+2[/latex] , [latex]f(x)=x^2-3[/latex] etc. they all give you a derivative of [latex]f'(x)=2x[/latex]. So we can say that [latex]2x[/latex] has many antiderivatives.

In fact if [latex]f(x) = 2x[/latex] , there will be whole family of antiderivatives, represented by [latex]F(x) = x^2+c[/latex],  where [latex]c\in\mathbb{R}[/latex], (that is, [latex]c[/latex] could be any real number).

Integration

Integration has many applications and can be used to find the area under the graph of a function. The Fundamental Theorem of Calculus shows us that, for practical purposes, the integral of a function is equivalent to the antiderivative of that function.

For example,  [latex]\displaystyle\int2x[/latex] [latex]{dx}=x^2+c[/latex]  where [latex]c[/latex] is some constant.

Note that the integral is written with the integral sign "[latex]\displaystyle\int[/latex]" before the function and the "[latex]{dx}[/latex] " after the function to show that it is the variable [latex]x[/latex] that is to be integrated. So the expression [latex]\displaystyle\int 2x\, dx[/latex] would be read as "the integral of [latex]2x[/latex], with respect to [latex]x[/latex]"

The process of integration can be more difficult than differentiation, but we have rules that we can use to help us find the integral.

Do you need this chapter?

Below is a quiz on Integration. If you can answer all these questions, then you can skip this chapter as you already have the requisite knowledge.

If you need to review this topic, continue reading.

The First rule of integration

The first rule is for the integration of polynomial functions.

\begin{align*}\int x^n{dx}&=\dfrac{1}{n+1}x^{n+1}+c \quad n\neq-1\\
&=\dfrac{x^{n+1}}{n+1}+c \quad n\neq-1\\\end{align*}

where [latex]c[/latex] can be any real number (that is, [latex]c\in\mathbb{R}[/latex]).

In fact, this rule will work for any power of [latex]x[/latex] except for [latex]-1[/latex].

Take a look at the examples below and then try solving the questions in the exercise below.

Note that each of these examples is using the rule [latex]\displaystyle \int x^n[/latex] [latex]{dx}=\dfrac{1}{n+1}x^{n+1}+c[/latex]

Note also that when a term is multiplied by a constant (a number), you can take that number out of the integral.

Also, when a number of terms are added or subtracted, you can integrate each term individually.

Now try the exercise below.

This rule: [latex]\displaystyle \int x^n[/latex] [latex]{dx}=\dfrac{1}{n+1}x^{n+1}+c[/latex]  can also be applied to negative and fractional powers (as long as [latex]n≠-1[/latex]).

Make sure that you know your index laws when dealing with negative or fractional indices. If necessary, refer to the index laws in the chapter on indices.

Examine the examples below and then complete the exercise.

Rules of integration

There are many other types of functions, besides those that have powers of [latex]x[/latex], such as trigonometric and exponential functions. We have many rules that can be used to help us integrate these functions.

Some of these rules are listed in the table below.

Examine the examples below and then use the formulas in the table to help you to complete the exercises.

When we are differentiating functions, we often need to use the Product Rule or the Quotient Rule or the Chain Rule. These rules do not apply for antidifferentiation (integration), but there are certain techniques such as integration by substitution, integration by parts and integration using partial fractions, that can be used to integrate more complex functions. These techniques are not covered in this book, but may be taught in your engineering or science program.

Definite integrals

In all of the examples so far, the integral of a function, gives us another function. These are known as "indefinite integrals".

It is also possible to evaluate an integral and find a specific value. This is known as a "definite integral".

For a function [latex]f(x)[/latex] suppose we can write an antiderivative [latex]\displaystyle\int f(x)[/latex] [latex]dx[/latex][latex]=F(x)[/latex]. We then define a definite integral of a function [latex]f(x)[/latex] from [latex]a[/latex] to [latex]b[/latex] with respect to [latex]x[/latex] using the notation:

\begin{align*}
\int_{a}^{b}f \left(x \right)dx&=\left[F\left(x \right)\right]_{a}^{b}\\
&=F\left(b \right)-F\left(a \right)\\
\end{align*}

where we assume [latex]f(x)[/latex] is defined for all [latex]x[/latex] in the interval [latex][a,b][/latex]. We call [latex]a[/latex] the lower limit of integration and [latex]b[/latex] the upper limit of integration. The notation [latex]\left[F(x)\right]_{a}^{b}[/latex] means substitute [latex]x=b[/latex] and [latex]x=a[/latex] into [latex]F(x)[/latex] and then subtract the values. (This eliminates the constant of integration [latex]c[/latex]).

Now try this exercise.