\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Integrals and Integration

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Integrals and Integration

The idea of integration is that integration 'undoes' the derivative, i.e. integration is the 'inverse' of differentiation. I used quotes around the word 'inverse' because in actuality differentiation and integration are mostly inverses but not completely. Let's start with what an integral is and some notation first.

Topics You Need To Understand For This Page

derivatives

If you have already studied integration and just want some practice problems, there are several places that you can find lots of practice problems on our site. This page contains basic practice problems for first semester calculus for students who are just starting to learn integration.

If you are in first semester calculus and you have learned all or most of the techniques you need for the course, our Calculus 1 Practice Integrals page will help you prepare for your integrals exam and your final exam. Those problems are mixed just as you will find on your exams.

If you are in second semester calculus and you have learned all or most of the advanced integrals, our Calculus 2 Practice Integrals page will help you prepare for your integrals exam and your final exam. Those problems are mixed just as you will find on your exams.

What Is An Integral?

Before we get to the details of integration, let's watch a quick video on exactly what integrals are. The material in this video is covered on several 17calculus pages but the video gives you a good overview of integration, what it means and some of the notation.

Physical Chemistry - What is an integral? [7min-21secs]

As you know, if you have a function, \(g(x)\), the derivative of \(g(x)\) is written
\(\displaystyle{ \frac{d}{dx}[g(x)] = G(x) }\)
In this equation, we are using \(G(x)\) to represent the new function we get after taking the derivative. When integrating, the notation looks like this
\(\int{F(x)~dx} = f(x) + C \)
In this equation, \(F(x)\) is the function we are integrating and \(f(x)+C\) is the result. The curved vertical line \(\int{}\) and the \(dx\) are both necessary in this notation. They act like brackets to indicate what is being integrated, which we call the integrand.

Okay, so what are integrals and how do we go about calculating them? Here is a great introduction video to integrals, what they represent and the notation. This is one of the best introductory videos you will find anywhere on any topic. So your time will be well spent watching this.

Krista King Math - Integrals - Calculus [ excellent video ] [5min-55secs]

video by Krista King Math


You will notice in the equation \(f(x)+C\), we have \(+C\) at the end. We need this because the derivative of a constant is zero. When we go in reverse, it is not possible to recover constants without more information. So, we use the \(+C\) as a placeholder for an unknown constant. Here is a video that explains this very well. If you are already confident with integrals, you can easily skip this video without losing anything. But, if you need a bit more explanation with some examples, this video may help.

Khan Academy - The Indefinite Integral or Anti-derivative [9min-28secs]

In this video he starts with the derivative \( \frac{d}{dx}[x^2] = 2x \).
Then explains how to work this backwards, i.e. given \(2x\) how do you get \(x^2\). Then he shows that the derivative of \( x^2+1\) is also \(2x\) and the derivative of \( x^2+2 \) is also \(2x\). So that the derivative of \( x^2 + any~constant\) is \( 2x\), usually written \(x^2+C\) where \(C\) just represents a constant.

Then he looks at the derivative of \(y=Ax^n\) which is \( (A \cdot n) x^{n-1} \) to try to figure out what \( y=\int{x^3 ~ dx} \) is.
This is a neat way of learning integration. He then generalizes the power rule for integration from this example.

Another example: \( \int{5x^7 dx} \) He is correct about not forgetting the \(+C\). Many instructors will take off points if you leave it off.

To sum up, although his explanation is a little choppy, this is a pretty good introduction video to integration.

video by Khan Academy

Some Basic Formulas

Here are a few basic formulas that you will need for upcoming pages. You already know the derivatives of these, so the anti-derivatives should contain no surprises.

\(\displaystyle{ \int{ k~dx } = kx + C }\)

 

\(\displaystyle{ \int{ x^r~dx } = \frac{x^{r+1}}{r+1} + C }\), \(r \neq -1\)

\(\displaystyle{ \int{ e^x~dx } = e^x + C }\)

 

\(\displaystyle{ \int{ \frac{1}{x} dx } = \ln(x) + C }\)

\(\displaystyle{ \int{ \sin(x) ~dx} = -\cos(x) + C }\)

 

\(\displaystyle{ \int{ \cos(x) ~dx } = \sin(x) + C }\)

Notes
1. In the above list, \(k\) is a constant and \(r\) is a rational number.
2. For the natural logarithm, be careful that you understand that \( \int{ \ln(x)~ dx } \neq 1/x + C \). This is a common mistake when students are first learning integration.
3. Watch the negative on the integral for sine.
4. There are comparable anti-derivatives for the other trig functions as well. You can find those details on the trig integration page.

Okay, time for some practice problems.

You Can Have an Amazing Memory: Learn Life-Changing Techniques and Tips from the Memory Maestro

Practice

For additional practice integrals, see the Calculus 1 Practice Integrals page.

Unless otherwise instructed, evaluate these integrals, giving your answers in exact form.

Basic

\( \int{ x^{2/3} ~dx } \)

Problem Statement

Evaluate the integral \( \int{ x^{2/3} ~dx } \). Give your answer in simplified, factored form.

Solution

Integrals ForYou - 118 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{3x^2+2x+1~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{3x^2+2x+1~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 996 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{3x^4+5x-6~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{3x^4+5x-6~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 997 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{1-2x^2+3x^3~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{1-2x^2+3x^3~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 998 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{\frac{-1}{x^2}dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{\frac{-1}{x^2}dx}}\), giving your answer in exact form.

Solution

Krista King Math - 999 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 1000 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 1001 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 1002 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{1+2x-4x^3~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{1+2x-4x^3~dx}}\), giving your answer in exact form.

Solution

PatrickJMT - 1005 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{(x+1)(x^2+3)~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{(x+1)(x^2+3)~dx}}\), giving your answer in exact form.

Solution

PatrickJMT - 1007 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\), giving your answer in exact form.

Solution

PatrickJMT - 1315 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\), giving your answer in exact form.

Solution

PatrickJMT - 1316 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Intermediate

\(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\), giving your answer in exact form.

Solution

Krista King Math - 1003 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\), giving your answer in exact form.

Solution

Krista King Math - 1004 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\)

Problem Statement

Evaluate \(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\), giving your answer in exact form.

Solution

PatrickJMT - 1006 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Calculus

Log in to rate this page and to see it's current rating.

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

math and science learning techniques

Shop Amazon - New Textbooks - Save up to 40%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

What Is An Integral?

Some Basic Formulas

Practice

next: definite integrals →

Practice Search
next: definite integrals →

Practice Instructions

Unless otherwise instructed, evaluate these integrals, giving your answers in exact form.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics