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AP Calculus AB & BC: Homework Help Resource17 chapters | 139 lessons
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Integrating various types of functions is not difficult. All you need to know are the rules that apply and how different functions integrate. You know the problem is an integration problem when you see the following symbol:
Remember, too, that your integration answer will always have a constant of integration, which means that you are going to add '+ C' for all your answers. The various types of functions you will most commonly see are monomials, reciprocals, exponentials, and trigonometric functions. Certain rules like the constant rule and the power rule will also help you. Let's start with monomials.
Monomials are functions that have only one term. Some monomials are just constants, while others also involve variables. None of the variables have powers that are fractions; all the powers are whole integers. For example, f(x) = 6 is a constant monomial, while f(x) = x is a monomial with a variable.
When you see a constant monomial as your function, the answer when you integrate is our constant multiplied by the variable, plus our constant of integration. For example, if our function is f(x) = 6, then our answer will be the following:
We can write this in formula form as the following:
If our function is a monomial with variables like f(x) = x, then we will need the aid of the power rule which tells us the following:
The power rule tells us that if our function is a monomial involving variables, then our answer will be the variable raised to the current power plus 1, divided by our current power plus 1, plus our constant of integration. This is only if our current power is not -1. For example, if our function is f(x) = x, where our current power is 1, then our answer will be this:
Recall that if you don't see a power, it is always 1 because anything raised to the first power is itself.
Let's try another example. If our function is f(x) = x^2, then our answer will be the following:
Whatever our current power is our answer will be the variable raised to the next power divided by the next power. In the above example, our current power is 2, so our next power is 3. In our answer, we have a 3 for the variable's power and for the denominator following the power rule. If our monomial is a combination of a constant and a variable, we have the constant rule to help us. The constant rule looks like this:
The constant rule tells us to move the constant out of the integral and then to integrate the rest of the function. For example, if our function is f(x) = 6x, then our integral and answer will be the following:
We've moved the 6 outside of the integral according to the constant rule, and then we integrated the x by itself using the power rule. For the answer, we simplified the 6x^2/2 to 3x^2 since 6x divides evenly by 2.
Another type of function we will deal with is the reciprocal. The integral of the reciprocal follows this formula:
The formula is telling us that when we integrate the reciprocal, the answer is the natural log of the absolute value of our variable plus our constant of integration. Exponential functions include the e^x function as well as the log(x) function and these types of functions follow these formulas for integration:
The first formula tells us that when we have a function e^x, our answer for the integral will be e^x + C. The a in the middle integral formula stands for a constant. The middle formula tells us that when we have, for example, a function like 3^x, then our answer after integrating will be 3^x/log(3) + C. The integral formula tells us that the integral of the natural log of x function is x(log(x) - 1) plus our constant of integration.
Our trigonometric functions include cosine, sine, and secant functions. They follow these formulas:
If you are integrating the cosine function, you will end up with the sine function plus the constant of integration. Integrating the sine function gives you the negative cosine function plus our constant of integration. If you see the secant function squared, your answer will be the tangent function plus our constant of integration.
Let's review. Integrating different functions involves referring to the formulas for each type of function along with applying the constant or power rule when necessary. The constant rule tells us to move the constant out of the integral and then to integrate the rest of the function. The power rule tells us that if our function is a monomial involving variables, then our answer will be the variable raised to the current power plus 1, divided by our current power plus 1, plus our constant of integration. Always remember your constant of integration when integrating.
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AP Calculus AB & BC: Homework Help Resource17 chapters | 139 lessons