Integration Using Trigonometric Identities

Sometimes, a trigonometric integral doesn’t suit integration by recognition. Trying to force a match with a known derivative can leave an extra factor (like a spare or ) that we can’t remove by dividing, since we’re only allowed to introduce or cancel constant factors. In these cases, we first use trigonometric identities to rewrite the integrand into an equivalent form that reveals a familiar derivative and then proceed by recognition (which we call "guesstimation").

What happens when we try to integrate a function such as using integration by recognition?

This is a power function, so suppose we guess . This can also be written as .

Differentiate our guess using the chain rule. This is a power function, so the angle is what’s inside the bracket, which is in this case. The angle differentiates to . Then, for the power function, we bring down the power, knock off the power and then keep the angle inside the same.

Angle stays the same



Differentiated angleDifferentiated power function

So, differentiates to give , which means integrates to give . We wanted to integrate , which would mean that we would have to divide both sides by to be left with the integral of .

This incorrect. We can only divide by constants, not by variables, and is a variable.

So, we need to use a trigonometric identity to rewrite as a function where we can use “guesstimation”.

Instead, using our double angle formulae, we can rewrite using . We rearrange this equation to make the subject of the equation.

Therefore,

Note: The constant can be taken outside the integral as it is a multiplier and makes adjusting the constants later much easier.

Now, let's make our guesses. We know integrates to give . For , we will guess . The angle in this case is , which differentiates to , differentiates to , and then the angle is kept the same.

Angle stays the same



Differentiated angle

Our guess differentiated to give , but we wanted , so we multiply both sides by to get . So, we are left with:

Factorising out , we get:

Note: When factorising out , we're essentially multiplying
every term in the bracket by .

General Steps on Integrating using Trigonometric Identities

1) Use a suitable trigonometric identity to rewrite our original function so that we can integrate by recognition (i.e. “guesstimation”).

2) Make a guess (or guesses if there’s more than one term) as to what may differentiate to give us the function we’re integrating.

3) Differentiate the guess.

4) If the derivative is a constant multiple of our function, adjust by multiplying the guess by , or we can see it as dividing the constant we want by the constant we have. Note that we can only do this with constants, not with variables.

5) Don’t forget to add a at the end (for indefinite integrals).