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Trigonometric Substitution

There is another class of integrals which usually do not involve trigonometric functions, but which can be solved by substituting the variable with a trigonometric function. This can be thought of as using the substitution formula, from Integration By Substitution, in the other direction. That is, going from the left side to right side in the equality

where we have made the substitution . We often use when making a trigonometric substitution.

Example

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Consider the substitution . Then one finds that . Making these substitutions and recalling the Pythagorean identity , the integral becomes

The last line comes from our original substitution:

Typical substitutions

Trigonometric substitution makes use of the Pythagorean identities. In general, the basic trigonometric substitutions are:

FormSubstitutionIdentity used

Caveat

The form often leads to a messy integral involving . This can often be avoided using a hyperbolic trigonometric substitution (see below).

After a substitution has been made, the resulting integral will often involve a product of trigonometric functions, possibly raised to powers. These types of integrals are covered in more detail in Trigonometric Integrals. For now, here are a few of the useful identities in evaluating these integrals:

Power reduction
Double angle

Example

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According to the above guide, the substitution to make is . Then , and it follows that

Now using the power reduction identity for cosine, we have

Finally, we must get this back in terms of . We know that . But to take care of , we must use the double angle formula from above. This gives

In the last line above, we knew from the original substitution. We found by drawing a right triangle which relates and according to the substitution :

Putting this all together and doing a little simplification, we find

Example

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By the above table, the substitution should be used (hence ). Then , so the integral becomes

Forms with other constants

There are other forms which are similar to the above forms but have different constants involved. These are dealt with using similar substitutions which make the constants cancel and factor so that the same identities can be used.

Example

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The form in the denominator reminds us of the substitution we made earlier for , which was the substitution . This is the correct impulse, but unfortunately it does not work quite right here since there is no nice simplification for .

We can fix this by adjusting the coefficients. The idea is that we could factor out a 4 if we had

To get that extra factor of 4, we can make the substitution . Then , and the integral becomes

The last equality above comes from again using the identity . Doing a little simplification and rewriting in terms of sine and cosine gives

This we can handle with a substitution of and , which gives

Now, we must do one final bit of right triangle trigonometry to get in terms of . By the original substitution we have , and this can be expressed by the following triangle:

It follows that . Putting it all together, we have

Example

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This is another example which looks like is the right type of substitution to make. However, again we need to adjust the coefficient since does not simplify nicely.

The key is to get the constants to cancel and factor. The substitution will work, and in this case . Then , and the integral becomes

The following table summarizes the substitutions to be made when other constants are involved. The identities used are the same Pythagorean identities given in the above table.

FormSubstitution

Completing the square

Sometimes it is not obvious at first that an integral is of the form where a trigonometric substitution is helpful. It may take a little bit of algebra to see what the right substitution is. This common algebraic tool is known as completing the square, which simply rewrites a quadratic expression as the square of a binomial plus a constant. To review the algebra involved in this process, check Wikipedia:Completing the square.

Example

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Start by completing the square for the quadratic:

So we can rewrite the integral as

where we substituted and . This can now be dealt with using a trigonometric substitution of (remember, the extra factor of 2 is there so that the 4 will factor out). So , and the integral becomes

Solving our original substitution for , we see that

So the final answer is

Hyperbolic trigonometric substitutions

Recall that the hyperbolic trigonometric functions and are defined by

These functions satisfy the Pythagorean identity . Also, note that , and . This means hyperbolic substitutions are another option for dealing with the following forms:

FormSubstitutionIdentity used

This often gives a simpler answer than the substitution suggested above, but the trade-off is that the answer will involve hyperbolic functions. Here are some of the other identities for the hyperbolic functions, which are similar to those for regular trigonometric functions:

Double Angle
Power reduction

Example

Compute

Using a regular trigonometric substitution, we would set , and , which, after the usual algebra, gives

But the integral of secant is not easy to remember, nor easy to rederive. If instead, we make the hyperbolic trigonometric substitution , so , then we have

So the hyperbolic trigonometric substitution led to a much easier integral to evaluate. The trade-off is that the final result involves the inverse hyperbolic trigonometric functions, as opposed to more familiar functions.

Example

Compute

Using the hyperbolic trigonometric substitution gives

Blow-ups

Sometimes a differential equation can be solved by using a trigonometric substitution. But this can sometimes lead to an unreasonable solution due to blow-ups or singularities, which exist for many trigonometric functions.

Example

Consider a financial model which predicts that marginal profits equal some positive constant plus something which is proportional to the square of net profits. Mathematically,

for constants and (we square them to ensure that they are positive). Solve this differential equation and find where it has a blow-up.

This is a separable equation. Separating the variables and integrating both sides gives

On the left, we can use the trigonometric substitution

Note then that . This gives

(leaving off the constant for now). On the right side we get , so

Solving this for gives

If initial profits, at , are 0, then , so the final answer is

Since tangent blows up at , this model implies profit goes to infinity at , which is a sign that this model is not perfect.


EXERCISES

Compute the following integrals: