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Integration Technique - Trigonometric Substitution |
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on this page: ► what to do with square roots ► the substitutions ► setting up integrals ► expanding this technique |
Difference Between Trig Integration and Trig Substitution
Trig integration is the evaluation of integrals that already have trig functions in the integrand.
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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. |
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 | |||
|---|---|---|---|
integrand | substitution | identity | |
\(x^2 + a^2\) |
\(x = a\tan(\theta)\) |
\( 1+\tan^2(\theta) = \sec^2(\theta)\) | |
\(-x^2 + a^2\) |
\(x = a\sin(\theta)\) |
\(\sin^2(\theta) + \cos^2(\theta) = 1\) | |
\(x^2-a^2\) |
\(x = a\sec(\theta)\) |
\( 1+\tan^2(\theta) = \sec^2(\theta)\) |
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a is a constant; x and θ are variables | |||
| table one - list of factors | |
|---|---|
a is a constant; x and θ are variables | |
integrand | \(x^2 + a^2\) |
substitution | \(x = a\tan(\theta)\) |
identity | \( 1+\tan^2(\theta) = \sec^2(\theta)\) |
integrand | \(-x^2 + a^2\) |
substitution | \(x = a\sin(\theta)\) |
identity | \(\sin^2(\theta) + \cos^2(\theta) = 1\) |
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 you use 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.
What To Do With Square Roots |
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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 | |
The Substitutions |
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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.
Tangent Substitution x2 + a2 → x = a tan(θ)
Overview
For this substitution, we also use the equation \(1+\tan^2(\theta) = \sec^2(\theta)\). Notice that when we perform the substitution, we have
\(\displaystyle{
\begin{array}{rcl}
a^2+x^2 & = & a^2 + (a\tan(\theta))^2 \\
& = & a^2 + a^2 \tan^2(\theta) \\
& = & a^2 (1 + \tan^2(\theta)) \\
& = & a^2 \sec^2(\theta) \\
\end{array}
}\)
In the last equation we end up with two squared terms, which we can easily take the square root of, for example, to simplify.
Triangle
As mentioned in step 2 above, we need to build a triangle based on our choice of substitution, in this case
\(x = a\tan(\theta) \to \tan(\theta) = x/a\). So, we draw a right triangle, select one of the other two angles to be \(\theta\) and label \(a\) and \(x\) accordingingly. Then we use the Pythagorean Theorem to get the expression for the other side. This gives us the triangle on the right.
Next, we substitute in the integral for \(x = a\tan(\theta))\) and \(dx = a\sec^2(\theta)~d\theta \). This last substitution is important to remember since we can't just replace \(dx\) with \(d\theta\) and it is a common source of error for many students.
Finally, once we are done integrating, we use the same triangle above to convert all the terms back into \(x\). It is not correct to leave \(\theta\)'s in the final answer.
Practice
Click here to go to a list of problems to practice this technique.
Sine Substitution – x2 + a2 → x = a sin(θ)
Overview
The other equation we use in this substitution is \(\sin^2(\theta) + \cos^2(\theta) = 1\). Using the substitution \( x = a\sin(\theta) \) in \(a^2-x^2 \) gives us
\(\displaystyle{
\begin{array}{rcl}
a^2-x^2 & = & a^2 - (a\sin(\theta))^2 \\
& = & a^2 - a^2 \sin^2(\theta) \\
& = & a^2(1-\sin^2(\theta)) \\
& = & a^2 \cos^2(\theta)
\end{array}
}\)
This may not seem like much but think about this. If the term \(a^2-x^2 \) is under a square root, then the last term above could be simplified quite a bit.
\(\sqrt{a^2-x^2 } = \sqrt{a^2 \cos^2(\theta)} = a\cos(\theta)\)
Notice we lost the square root and now have a term that, depending on it's location in the integrand, is much easier to integrate.
Triangle
As mentioned in the steps above, we need to draw a triangle based on our choice of substitution, in this case \( x = a\sin(\theta) \to \sin(\theta) = x/a \). Using this last equation, we draw a right triangle and label the angle \(\theta\), then sides \(x\) and \(a\). Then we use the Pythagorean Theorem to get the expression for the third side.
Next, we substitute in the integral for \(x = a\sin(\theta)\) and \(dx = a\cos(\theta)~d\theta \). This last substitution is important to remember since we can't just replace \(dx\) with \(d\theta\) and it is a common source of error for many students.
Finally, once we are done integrating, we use the same triangle above to convert all the terms back into \(x\). It is not correct to leave \(\theta\)'s in the final answer.
Practice
Click here to go to a list of problems to practice this technique.
Secant Substitution x2 – a2 → x = a sec(θ)
Overview
In this case, we have two choices, to use secant or cosecant. Fortunately, both should work. So you may choose either based on what you are comfortable with, what your instructor requires and any other restraining factors. We will discuss the use of cosecant but the use of secant parallels this and we think you can easily extrapolate and alter the discussion to fit secant.
Similar to the other two cases above, we have the equation \(1+\cot^2(\theta) = \csc^2(\theta)\) that we will use to simplify. Performing the substitution, we have
\(\displaystyle{
\begin{array}{rcl}
-a^2+x^2 & = & -a^2 + (a\csc(\theta))^2 \\
& = & -a^2 + a^2 \csc^2(\theta) \\
& = & a^2(-1+\csc^2(\theta)) \\
& = & a^2 \cot^2(\theta)
\end{array}
}\)
This is a much simpler expression, especially if the \(-a^2+x^2\) is under a square root.
Triangle
We use \(x = a~\csc(\theta) \to \csc(\theta) = x/a \to 1/\sin(\theta) = x/a \to \sin(\theta) = a/x \) to build the triangle on the right. We choose one angle to label \(\theta\), label the other two sides using the last equation and then calculate the expression for the third side using the Pythagorean Theorem.
Next, we substitute in the integral for \(x = a\csc(\theta)\) and \(dx = a\csc(\theta)\cot(\theta)~d\theta \). This last substitution is important to remember since we can't just replace \(dx\) with \(d\theta\) and it is a common source of error for many students.
Finally, once we are done integrating, we use the same triangle above to convert all the terms back into \(x\). It is not correct to leave \(\theta\)'s in the final answer.
Practice
Click here to go to a list of problems to practice this technique.
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] | |
Setting Up Integrals Using Trig Substitution |
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Okay, now that you know what substitution to use, let's learn how to use those substitutions. Here is a great video showing how to set up each of the three types of problems discussed above.
Krista King Math - Setting Up Trigonometric Substitution | |
Expanding The Use of This Technique |
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In the discussion so far on this page, we have assumed that the \(x\) variable term is always exactly \(x\). There is a generalization of this where you might have \([f(x)]^2 + a^2\) and \(f(x) \neq x\). This technique still works and you just change the formulas as follows.
| table two | ||
|---|---|---|
integrand | substitution | |
tangent | \([f(x)]^2+a^2\) | \(f(x) = a\tan(\theta)\) |
sine | \(-[f(x)]^2+a^2\) | \(f(x) = a\sin(\theta)\) |
secant | \([f(x)]^2-a^2\) | \(f(x) = a\sec(\theta)\) |
f(x) can be pretty much anything (but not a constant, so check that it contains at least one x). Some examples are
| table three | |
|---|---|
integrand | substitution |
\(e^{2x}+9\) | \(e^x = 3\tan(\theta)\) |
\(-(x+2)^2+16\) | \(x+2 = 4\sin(\theta)\) |
\(4x^2-25\) | \(2x = 5\sec(\theta)\) |
Search 17Calculus
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Practice Problems |
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Instructions - - Unless otherwise instructed, evaluate the following integrals. Give all answers in exact, simplified form.
Level A - Basic |
Practice A04 | |
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\(\displaystyle{ \int_{10\sqrt{3}}^{10}{\frac{1}{\sqrt{x^2-64}}~dx } }\) | |
answer |
solution |
Practice A09 | |
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\(\displaystyle{ \int{ \frac{dx}{(a^2+x^2)^{3/2}}} }\) | |
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solution |
Practice A10 | |
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\(\displaystyle{ \int_{0}^{3}{\sqrt{9-x^2}~dx} }\) | |
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solution |
Level B - Intermediate |
Practice B01 | |
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Solve the initial value problem \(\displaystyle{\frac{dy}{dt}=\frac{1}{25+9t^2},~~~y(5/3)=\frac{\pi}{30}}\) | |
answer |
solution |
Practice B09 | |
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\(\displaystyle{ \int{ \frac{dx}{x^2+4x+1} } }\) | |
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solution |
Level C - Advanced |
