- Understand and apply the net change theorem to calculate how quantities change over an interval
- Use integration formulas to calculate the integrals of odd and even functions
In this section, we will use basic integration formulas to solve key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. While definite and indefinite integrals are closely related, there are some key differences:
- A definite integral represents a number (when the limits of integration are constants) or a function (when the limits are variables).
- An indefinite integral represents a family of functions, all differing by a constant.
As you become more familiar with integration, you will learn when to use definite integrals and when to use indefinite integrals. You will naturally select the correct approach for a given problem without much thought. However, until these concepts are firmly understood, consider carefully whether you need a definite or indefinite integral and use the proper notation accordingly.
Basic Integration Formulas
To solve problems using integration, we need to recall the integration formulas given in the Table of Antiderivatives (below) and the properties of definite integrals covered in the Differentiation Rules section.
| Differentiation Formula | Indefinite Integral |
|---|---|
| [latex]\frac{d}{dx}(k)=0[/latex] | [latex]\displaystyle\int kdx=\displaystyle\int kx^0 dx=kx+C[/latex] |
| [latex]\frac{d}{dx}(x^n)=nx^{n-1}[/latex] | [latex]\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}+C[/latex] for [latex]n\ne −1[/latex] |
| [latex]\frac{d}{dx}(\ln |x|)=\frac{1}{x}[/latex] | [latex]\displaystyle\int \frac{1}{x}dx=\ln |x|+C[/latex] |
| [latex]\frac{d}{dx}(e^x)=e^x[/latex] | [latex]\displaystyle\int e^x dx=e^x+C[/latex] |
| [latex]\frac{d}{dx}(\sin x)= \cos x[/latex] | [latex]\displaystyle\int \cos x dx= \sin x+C[/latex] |
| [latex]\frac{d}{dx}(\cos x)=− \sin x[/latex] | [latex]\displaystyle\int \sin x dx=− \cos x+C[/latex] |
| [latex]\frac{d}{dx}(\tan x)= \sec^2 x[/latex] | [latex]\displaystyle\int \sec^2 x dx= \tan x+C[/latex] |
| [latex]\frac{d}{dx}(\csc x)=−\csc x \cot x[/latex] | [latex]\displaystyle\int \csc x \cot x dx=−\csc x+C[/latex] |
| [latex]\frac{d}{dx}(\sec x)= \sec x \tan x[/latex] | [latex]\displaystyle\int \sec x \tan x dx= \sec x+C[/latex] |
| [latex]\frac{d}{dx}(\cot x)=−\csc^2 x[/latex] | [latex]\displaystyle\int \csc^2 x dx=−\cot x+C[/latex] |
| [latex]\frac{d}{dx}( \sin^{-1} x)=\frac{1}{\sqrt{1-x^2}}[/latex] | [latex]\displaystyle\int \frac{1}{\sqrt{1-x^2}} dx= \sin^{-1} x+C[/latex] |
| [latex]\frac{d}{dx}(\tan^{-1} x)=\frac{1}{1+x^2}[/latex] | [latex]\displaystyle\int \frac{1}{1+x^2} dx= \tan^{-1} x+C[/latex] |
| [latex]\frac{d}{dx}(\sec^{-1} |x|)=\frac{1}{x\sqrt{x^2-1}}[/latex] | [latex]\displaystyle\int \frac{1}{x\sqrt{x^2-1}} dx= \sec^{-1} |x|+C[/latex] |
Let’s look at a few examples of how to apply these rules.
Use the power rule to integrate the function [latex]{\displaystyle\int }_{1}^{4}\sqrt{t}(1+t)dt.[/latex]
Find the definite integral of [latex]f(x)={x}^{2}-3x[/latex] over the interval [latex]\left[1,3\right].[/latex]