1 Integration as differentiation in reverse

Suppose we differentiate the function $y=x^2$ . We obtain $\frac{dy}{dx}=2x$ . Integration reverses this process and we say that the integral of $2x$ is $x^2$ . Pictorially we can regard this as shown in Figure 1:

Figure 1

The situation is just a little more complicated because there are lots of functions we can differentiate to give $2x$ . Here are some of them: $x^2+4,x^2-15,x^2+0.5$

All these functions have the same derivative, $2x$ , because when we differentiate the constant term we obtain zero. Consequently, when we reverse the process, we have no idea what the original constant term might have been. So we include in our answer an unknown constant, $c$ say, called the constant of integration . We state that the integral of $2x$ is $x^2+c$ .

When we want to differentiate a function, $y\left(x\right)$ , we use the notation $\frac{d}{dx}$ as an instruction to differentiate, and write $\frac{d}{dx}\left(y\left(x\right)\right)$ . In a similar way, when we want to integrate a function we use a special notation: $\int \; <!--nolimits\; -->y\left(x\right)dx$ .

The symbol for integration, $\int \; <!--nolimits\; -->$ , is known as an integral sign . To integrate $2x$ we write

Note that along with the integral sign there is a term of the form $dx$ , which must always be written, and which indicates the variable involved, in this case $x$ . We say that $2x$ is being integrated with respect to $x$ . The function being integrated is called the integrand . Technically, integrals of this sort are called indefinite integrals , to distinguish them from definite integrals which are dealt with subsequently. When you find an indefinite integral your answer should always contain a constant of integration.

Exercises

1. 1. Write down the derivatives of each of:   $x^3,x^3+17,x^3-21$

        

   2. Deduce that $\int \; <!--nolimits\; -->3x^{2}dx=x^3+c.$
2. Explain why, when finding an indefinite integral, a constant of integration is always needed.

Answer