Business Calculus with Excel

In the last section we looked at the fundamental theorem of calculus and saw that it could be used to find definite integrals. We saw:

We thus find it very useful to be able to systematically find an anti-derivative of a function. The standard notation is to use an integral sign without the limits of integration to denote the general anti-derivative. Thus, ${\int }_a^{b}f(t)dt$ is referred to as the definite integral of $f(x)$ from $a$ to $b\text{,}$ and it is a number. In contrast, $\int f(x)dx$ is the indefinite integral of $f(x)\text{,}$ and it is a function. We use indefinite integrals or anti-derivatives to evaluate definite integrals or areas.

We find anti-derivatives by starting with the differentiation formulas of basic functions and manipulating them so the derivative is a nice function.

Elementary Anti-derivative 1 — Find a formula for $\int x^{n}dx\text{.}$

We start with the closest differentiation formula $\frac{d}{dx}{x}^n=n{x}^{n-1}\text{,}$ and manipulate it so $x^n$ is on the right hand side. We first replace $n$ with $n+1$ to get $\frac{d}{dx}{x}^{n+1}=(n+1)x^n\text{.}$ We then divide both sides by $n+1$ to obtain $x^n=\frac{d}{dx}{x}^{n+1}/(n+1)\text{.}$ Finally, we note that adding a constant $C$ does not change the derivative, so $x^n=\frac{d}{dx}(x^{n+1}/(n+1)+C)\text{.}$ Since we have divided by $n+1\text{,}$ we need to insist that $n+1\ne 0\text{.}$ Using the notation of indefinite integrals we obtain our power rule formula:

Note that this matches the pattern we found in the last section.

Elementary Anti-derivative 2 — Find a formula for $\int 1/xdx\text{.}$

We start with the closest differentiation formula $\frac{d}{dx}\ln (x)=1/x\text{.}$ In this case, we need to note that natural logarithms are only defined positive numbers and we would like a formula that is true for positive and negative numbers. We can do this with an appropriate use of absolute value bars. Thus, $\frac{d}{dx}(\ln (|x|)+C)=1/x\text{,}$ and we have our second formula:

Elementary Anti-derivative 3 — Find a formula for $\int e^{x}dx\text{.}$

Once again, we start with the closest differentiation formula $\frac{d}{dx}{e}^x=e^x\text{.}$ In this case we don’t have to do any manipulation, and we have our formula:

Elementary Anti-derivative 4 — Find a formula for $\int a^{x}dx$ for a positive number $a\text{.}$

This formula requires a bit more work. We start with the formula $\frac{d}{dx}{a}^x=\ln (a)a^x\text{.}$ Dividing both sides by the constant $\ln (a)$ gives $a^x=\frac{d}{dx}(a^x/\ln (a)+C)\text{.}$ Thus our integral is:

Sum, Difference, and Constant Multiple rules — The rules we had for taking derivatives of sums, differences, and constant multiples of functions translate into similar rules for integrals.

The derivatives of a sum rule, $\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)\text{,}$ becomes the

The derivatives of a difference rule, $\frac{d}{dx}(f(x)-g(x))=\frac{d}{dx}f(x)-\frac{d}{dx}g(x)\text{,}$ becomes the

We can use these rules to find the indefinite integrals on a lot of functions. They cover all polynomials.

We can also use these rules to find indefinite integrals for roots.

We can also find anti-derivatives of exponential and power functions.

As we mentioned earlier in the section, the normal reason for wanting to find indefinite integrals is to be able to use them with the fundamental theorem of calculus to find definite integrals.

Another application for anti-derivatives is solving an initial value problem. In that case we want to a particular anti-derivative that has a particular value for a specified $x\text{.}$ In this situation we may not set $C$ to zero. In fact, part of the problem will be to find the appropriate value of $C\text{.}$

A word of warning — The anti-differentiation formulas we have produced only work for the functions given, allowing for changes in variables. At this point the only way we have for finding $\int (3x+5{)}^{2}dx$ is expand the integrand getting $\int (9x^2+30x+25)dx$ before applying our rules. In general, the process of finding anti-derivatives symbolically is an art form that we only begin to work with in this course.