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

17Calculus Integrals - Trig Substitution

Limits

Using Limits

Limits FAQs

Derivatives

Graphing

Related Rates

Optimization

Other Applications

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

Infinite Series

Applications

Tools

Parametrics

Conics

Polar Coordinates

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

Difference Between Trig Integration and Trig Substitution

Trig integration is the evaluation of integrals that already have trig functions in the integrand.

Trig substitution, covered on this page, is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

This is a great technique that introduces trig into an integral which originally doesn't have any trig. This makes integration easier and sometimes possible where it wasn't before.

If you want a full lecture on trig substitution, we recommend this video from one of our favorite lecturers.

Prof Leonard - Calculus 2 Lecture 7.3: Integrals By Trigonometric Substitution [2hrs-9min-23secs]

video by Prof Leonard

The key to using this technique is to recognize specific sets of terms in the integrand, which will tell you the associated substitution. Table one contains a summary.

Table One - List of Factors

<\(a\) is a constant; \(x\) and \(\theta\) are variables

tangent

integrand

\(x^2 + a^2\)

substitution

\(x = a\tan(\theta)\)

identity

\( 1+\tan^2(\theta) = \sec^2(\theta)\)

sine

integrand

\(-x^2 + a^2\)

substitution

\(x = a\sin(\theta)\)

identity

\(\sin^2(\theta) + \cos^2(\theta) = 1\)

secant

integrand

\(x^2-a^2\)

substitution

\(x = a\sec(\theta)\)

identity

\( 1+\tan^2(\theta) = \sec^2(\theta)\)

Note - - Some books and instructors may use cosecant substitution instead of secant. In this case, the substitution is \(x = a\csc(\theta)\) and the identity is \( 1+\cot^2(\theta) = \csc^2(\theta) \).

The terms you are looking for in the integrand may appear exactly as they are listed in table one or, more likely, they will be embedded in the integrand, like under a square root or with a power. They can appear in the numerator or the denominator of a fraction. Sometimes you may even have to complete the square in order to see the term in this form. In any case, once you have recognized the form of the term, you can choose the correct the substitution. Here are the steps to evaluate integrals using this technique.

1. Choose a substitution from table one depending on the term that appears in the integrand.
2. Draw a triangle based on the substitution.
3. Perform the substitution in the integral, simplify and evaluate.
4. Use the triangle from step 2 to convert your answer back to the original variable.

Definite Integrals
There is one possible variation to these steps. If your integral is a definite integral, you can follow the steps above and then finish the problem by evaluating the result at the limits of integration. Or you could convert your limits of integration using the substitution you chose in step 1 and then evaluate them after step 3 and skip step 4. This is up to you based on the problem statement, what you find the easiest to do and on what your instructor expects. If you are not sure, ask your instructor.

Setting Up Integrals Using Trig Substitution

Okay, before we get into the details of this technique, we need to know how to set up the integrals. Here is a great video showing how to set up each of the three types of problems.

Krista King Math - Setting Up Trigonometric Substitution [13min-12secs]

video by Krista King Math

What To Do With Square Roots

So, what do you do when you have one of the factors in table one under a square root sign? Well, you do the same substitution as if the square root sign was not there. To understand this better, here is a video explaining this and showing an example.

Dr Chris Tisdell - Integrals With Square Root Signs [15min-33secs]

video by Dr Chris Tisdell

Before going on, here is a quick video overview of the detail just discussed. This video may help fill in any gaps in your understanding of the basics of this technique.

Michel vanBiezen - Trig Substitution - What Is & When to Use Trig Substitution? [2min-57secs]

video by Michel vanBiezen

Now, let's look at each substitution separately. The steps for each case are pretty much the same but we discuss each case separately to help you understand this technique. Your next logical step is to look at the tangent substitution on a separate page.

You CAN Ace Calculus

Topics You Need To Understand For This Page

Related Topics and Links

external links you may find helpful

trig substitution youtube playlist

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

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

 

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

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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

Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

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.

more calculus help

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Learn to Remember : Practical Techniques and Exercises to Improve Your Memory

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Prime Student 6-month Trial

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Kindle Unlimited Membership Plans

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 © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.