Integration Problems in Calculus: Solutions, Examples & Quiz

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Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught Math at a public charter high school.

In this lesson, learn about the different types of integration problems you will encounter. You will see how to solve each type. Also, learn about the rules of integration that will help you.

We also recommend watching The Fundamental Theorem of Calculus and Using Integration By Parts

Integration Problems

First, let me say that 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.

The integral symbol.
integration problems

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

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.

Integrating a constant monomial.
integration problems

We can write this in formula form as the following.

The formula for integrating a constant monomial.
integration problems

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.
integration problems

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 one, 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.

Integrating the function f(x) = x using the power rule.
integration problems

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.

Integrating the function f(x) = x^2 using the power rule.
integration problems

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.
integration problems

This 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.

Integrating the function f(x) = 6x using both the constant rule and the power rule.
integration problems

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 6 divides evenly by 2.

Reciprocals and Exponentials

Another type of function we will deal with is the reciprocal. The integral of the reciprocal follows this formula.

The formula for integrating the reciprocal function.
integration problems

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