Two ways to use trigonometry in integrals

We use trigonometric functions quite often in integration, even when there are no trig. functions explicitly in the integral. In this section we'll look at
  1. Trigonometric integrals and
  2. Integration by substitution of trig. functions (or "trig. substitution").

Trigonometric Integrals

Working with just a few key trigonometric identities can open up a whole range of trigonometric integrals for you. For example, it's very important to be able to do integrals like

equation 1 large

because they arise in an important application of mathematics called Fourier Analysis. Fourier integrals are important in physics, chemistry, medical imaging, communications, noise filtering and many other fields.

You couldn't get an MRI (magnetic resonance imaging) for your aching knee if we didn't have Fourier analysis.

Important trig. identities

First we recall the Pythagorean identity:

pythagorean identity.

If we begin with the cosine double angle formula,

cos double angle formula

we can use the Pythagorean identity to substitute   1 - cos2θ   for sin2θ to obtain one of the power-reduction identities:

power reduction 1

Notice that this identity allows us down-convert the power of the cosine function from 2 to 1. And it's easy to integrate a function like cos(2θ) or sin(2θ) by simple substitution.

This will be very valuable in doing trig. integrals. Likewise, we can develop a sine-power conversion formula:

sine power conversion formula

In a similar way, we can take the Pythagorean identity for secant and tangent,

secant tangent pythag ident

(obtained by dividing the Pythagorean identity through by cos2θ), to obtain the tangent power-reduction formula:

tan power reduction formula

One use of trigonometric identities in integrals is to reduce the power (exponent) of one part of the integrand by one.

identity table

Summary of the key identities

← Here are the main identities to keep in mind when doing integrals of trigonometric functions.

Now we'll use them to solve integrals of the type

int sin^m cos^n

These will be divided into two categories below, those where at least one of m or n are odd, and integrals where both m or n are even. The best way to do this is by example, so here we go ...


Example 1 – Odd powers only   sin3x dx

To do this integral, regognize that sin3x = sin(x)·sin2(x), and write the new integral:

example 1 equation 2

Now use the identity

example 1 equation 3

to replace sin2x and write the new integral

example 1 equation 4

Now this new integral is a sum of two integrals, the last of which can be evaluated easily using the substitution u = cos(x), like this:

example 1 equation 5

The first integral is easy, it's just -cos(x). The second is easy because of the substitution.

example 1 equation 6

Now we just back substitute cos(x) for u to get the solution (don't forget the constant).

example 1 equation 7

We were able to do this integral with just simple substitution of trig. identities because the coefficient of the sine function was odd. In order for simple substitution like this to work, at least one of the exponents of the trig. functions has to be odd.


Example 2     tan3x dx

Find the indefinite integral:    

This is an example very similar to the first, but it will be a little more complicated to integrate. First use the Pythagorean identity for tangent and secant

example 2 equation 2

From this we get two integrals, each of which can be solved by simple substitution (u-substitution). Here's what it looks like:

example 2 equation 3

Putting it all together (and remembering that the integral on the right had a minus sign in front of it), we get the result:

example 2 equation 4

 


Example 3 – A mixed sine-cosine integral   sin3x cos2x dx

Using the Pythagorean identity, we can take advantage of the odd power of the sine part to reduce it to sin(x), which will come in handy later.

example 3 equation 2

Expand the integral to get the integral of a sum.

example 3 equation 4

Each of these is now easy to do with this simple substitution:

example 3 equation 5

to get these integrals:

example 3 equation 6

Evaluation of these and re-substitution of cos(x) for u gives the final integral:

example 3 equation 7.

When an odd power of a trig. function is present in an integrand, the object is to convert it to a product containing the first power of that function. That first power will come in handy later when integrating by u-substitution.


Example 4 – Larger powers   sin4x cos5x dx

First substitute (1 - sin2x)2 for cos4x to leave a factor of cos(x) behind. That bit will be useful in our u-substitution approach later.

example 4 equation 2

Now continue to expand the integrand (I'm a poet and I don't even realize it!), to get three integrals:

example 4 equation 3

These can all be solved using the u-substitution, Let u = sin(x) & du = cos(x) dx to get the string of easy to solve polynomial integrals:

example 4 equation 5

Then back-substituting sin(x) for u, we arrive at the solution:

example 4 equation 6

You can differentiate this result to see that it gets you right back to the original integrand.


Example 5 – A mixed tan|sec integrand   tan3x sec2x dx

By now I hope you're getting the hang of these. Go for the odd power on the tangent by replacing tan2(x) with sec2(x) - 1.

example 5 equation 2

Now expand to get the integral of a sum (which is the sum of integrals):

example 5 equation 3

Then make a substitution,

example 5 equation 4

and solve the easy power integrals:

example 5 equation 5

Finally, back substitute to get the solution.

example 5 equation 6


Example 6 – Solving integrals of the form  sinmx cosnx dx when both m and n are even

Evaluate the integral   sin2x dx 

For integrals with only even powers of trigonometric functions, we use the power-reduction formulae,

example 6 equation2,

to make the simple substitution

example 6 equation 3

Then we can separate this integral of a sum into the sum of integrals. The first is trivial, and the second can be don by u-substitution.

example 6 equation 4

example 6 equation 5

Both integrals are easy now (the first is already done below).

example 6 equation 6

We then take the result,

example 6 equation 7,

and back substitute u = 2x to arrive at the solution to the integral:

example 6 equation 8


Example 7 – Even exponents:   sin2x cos2x dx

First we use our sine and cosine power-reduction formulas in this simple substitution:

example 7 equation 2

to get a new integral:

example 7 equation 3

Now this integral contains a cos2 term. We can use the power reduction formula again here, this time with 2x substituted for x:

example 7 equation 4

Make the substitution and carry out the integration. To integrate the last term, we use the u-substitution u = 4x, then du = 4dx. Be careful with all of the various fractional factors that result.

example 7 equation 5

Finally, the integral is

example 7 equation 6

 

Why do we need to learn to do trig. integrals like this by hand? One important reason is that they are crucial in understanding the Fourier transform, something you might encounter later in math. The FT is extremely important in many aspects of modern life. You couldn't get an MRI on a sore joint without the Fourier transform, a way of transforming between the time and frequency domains in any study of electromagnetic radiation. There are other uses for trig. integrals, including the study of noise reduction and signal processing (like sending data), but the FT is a big one.


Trig. substitution in radical expressions

So far we've solved trigonometric integrals using trig. identity substitution and a few other small tricks. Now let's substitute some trigonometric functions for algebraic variables in algebraic expressions like these (a is a constant):

alg sub equation 1

This will be a nifty trick to solve some integrals you probably haven't cracked yet.

Let's start with the first radical expression, (a2 - x2)1/2. We begin by introducing a dummy variable, t, and making the substitution x = a sin(t). Then

alg sub equation 2

The substitution has reduced a radical to a simple trigonometric expression, the integral of which we know, so there's hope for this kind of substitution.

In a similar way we can substitute x = a tan(t) for the x in the second radical and x = a sec(t) for the x in the third. Each substitution leads to a simple trigonometric function.

See the table below for a summary of integration by trig. substitution.

 

 

Example 8:   x = a·sin(t) substitution

Integrate:

To begin, consult the table above and make the substitution x = a sin(t), where a = 9 (the square root of 81):

Substitute and simplify the expression under the radical.

That last step uses the Pythagorean identity, sin2t + cos2t = 1. Now the integrand, including dx, can be written:

The substitution allows us to take the square root:

Now move the constant out and use our cosine power-reduction formula to change the cos2(t) term into something we can integrate, just as in the examples above. Integrate the cos(2t) integrand with simple u-substitution.

The integral in terms of the dummy variable t is now easy to write, and we can substitute expressions containing x for t and sin(t). We also remember here that sin(2x) = 2 sin(x) cos(x) is another of our double-angle identities:

Now let's go back to the definition of t that we started with, namely that sin(t) = x/9, and construct the right triangle (angles not to scale) that goes with it. Note that we use the Pythagorean theorem to get the length of the lower side of the triangle.

That gives us cos(t), so we can plug sin(t) and cos(t) into the previous result to get

Now just some easy cancellation gives us the final integral:

 


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