Integration using the Power Rule

Introduction

We will begin our mastery of integration by understanding one of the most used rules, the power rule. Before we proceed to the illustrative examples, let us recall some basic integration rules:

The idea of the constant rule is to extract any constants, if possible, outside of the integrand, making it much simpler to apply the requires integration techniques.

As long as there are functions that are separated by either addition or subtraction, you can rewrite them with their respective integrals.

Similar to the constant rule, you can extract constants outside the integrand, to make integration easier.

To apply the power rule, you need to make sure that the integrand is in the form of a variable raised to a numerical exponent. In the meantime, we exclude the case of having $-1$  as an exponent, which will be discussed in the next lesson. Let's integrate!

Illustrative Example #1

Evaluate the integral \displaystyle \int 15x^3 \, dx.

First, we will use the constant multiple rules to extract 15 outside of the integrand, resulting in:

Since the integrand is  a variable raised to a numerical exponent, we can now apply the power rule: (Don't forget to add +C!)

Illustrative Example #2

Evaluate the integral \displaystyle \int -30x^5 \space dx.

First, we will use the constant multiple rules to extract -30 outside of the integrand, resulting in:

Since the integrand is  a variable raised to a numerical exponent, we can apply the power rule: (Don't forget to add +C!)

Illustrative Example #3

Evaluate the integral \displaystyle \int \bigl(-24x^5 - 10x\bigr) \, dx.

Illustrative Example #4

Evaluate the integral \displaystyle \int \bigl(\pi x + 5x^4\bigr) \, dx.

Illustrative Example #5

Evaluate the integral \displaystyle \intop 3x^{\frac{3}{2}} \, dx.

While it is convenient to use the power rule if the integrand is in the form of a variable raised to a numerical exponent, some integrals require rewriting the integrand applying the laws of radicals and exponents, and/or simplifying it by multiplication and division. The next examples are the following:

Illustrative Example #6

Evaluate the integral \displaystyle \int \sqrt[3]{x} \, dx.

Illustrative Example #7

Evaluate the integral \displaystyle \int \bigl(\sqrt[5]{x} + 2\sqrt[3]{x}\bigr) \, dx.

Illustrative Example #8

Evaluate the integral \displaystyle \int \bigl(x^3 - 5\bigr)\sqrt{x} \, dx.

Illustrative Example #9

Evaluate the integral \displaystyle \int \dfrac{x^4 - 1}{x^2} \, dx

We will perform division first in the integrand:

We then perform the integration using the Power Rule:

Illustrative Example #10

Evaluate the integral \displaystyle \int \frac{x^5 - 4x^3 + 2x}{x^3} \, dx.

We will perform division first in the integrand:

We then perform the integration using the Power Rule: