In this explainer, we will learn how to find indefinite integrals that result in trigonometric functions.

To find the indefinite integral of various trigonometric functions, we can start by recalling the first part of the fundamental theorem of calculus.

### Theorem: The Fundamental Theorem of Calculus

If is a continuous real-valued function on an interval and we let be defined on the interval as then is uniformly continuous on and differentiable on , and for all .

The fundamental theorem of calculus tells us that differentiation and indefinite integration are the reverse processes of one another. This means we can find the indefinite integral of some trigonometric functions by considering the derivative results we already know.

For example, we know that, for any real constant and variable measured in radians,

This tells us that is an antiderivative of , for any value of . We can use this to find the most general antiderivative of ; since this will be an antiderivative if we add any constant, we need to add a constant of integration we will call :

This is a useful result; however, we can rearrange this equation to find an integral rule for by using our properties of indefinite integrals:

This means

We can then divide through by , provided it is nonzero:

Finally, is a constant, which means is also just a constant. This means we can introduce a new constant to simplify this expression.

We have shown, provided , that

We can follow exactly the same process to find an indefinite integral rule for .

First, we know that, for any real constant and variable measured in radians,

This means is an antiderivative of , and we can use this to find the most general antiderivative:

We can take the constant factor of outside of our integral:

We can then divide through by , provided :

Finally, we set :

We will show one more example of this process on the tangent function.

We recall that, for any real constant and variable measured in radians,

This tells us that is an antiderivative of . Hence, we can use this to find the most general antiderivative:

Taking the constant factor outside of the indefinite integral gives us

Then, provided , we can divide through by : where .

Letβs summarize the indefinite integral results we have just shown.

### Definition: Indefinite Integrals Resulting in Trigonometric Functions

For any real constant and variable measured in radians,

- ,
- ,
- .

One way of visualizing this relationship is to use the following diagram.

Differentiating with respect to takes us clockwise through this cycle and integrating with respect to will take us counterclockwise, where we will need to add a constant of integration.

Letβs look at a few examples to practice and help strengthen our understanding. In our first example, we will demonstrate how to evaluate the indefinite integral of a sum of two trigonometric functions.

### Example 1: Integrating Trigonometric Functions

Determine

### Answer

We want to evaluate the indefinite integral of a trigonometric function. We can start by simplifying this integral using our properties of indefinite integration:

We can then evaluate these indefinite integrals by recalling

Combining all of the constants of integration at the end of our expression, this gives us

Hence,

In our next example, we will see how to apply this process to the indefinite integral of a trigonometric function involving multiple angles.

### Example 2: Integrating Trigonometric Functions with Multiple Angles

Determine

### Answer

To evaluate this indefinite integral, we start by recalling the following indefinite integral result for trigonometric functions. For any real constant ,

To apply this, we set and we take the constant factor 3 outside of our integral:

Finally, we will set and rewrite the factor of in front of our function, giving us

Hence,

We can also use these rules for indefinite integrals to evaluate integrals involving the antiderivative of the square of the secant function.

### Example 3: Integrating Reciprocal Trigonometric Functions

Determine

### Answer

To evaluate this indefinite integral, we first recall the following indefinite integral result for trigonometric functions. For any real constant ,

To apply this result to the indefinite integral in the question, we will first take the constant factor outside of our integral:

We then apply our indefinite integral rule with :

Finally, we will introduce a new constant and write the coefficient of at the front of the function:

Therefore, we have shown

We can also apply these results when the coefficient of our variable is not an integer. We will demonstrate this in our next example.

### Example 4: Integrating Trigonometric Functions

Determine

### Answer

To help us evaluate this indefinite integral, it is easier to rewrite the argument as :

We can then evaluate this integral by recalling the following indefinite integral rules for trigonometric functions; for any real constant ,

We can use this to evaluate our indefinite integral by taking the constant factor 7 outside the integral and setting . This gives us

Finally, we can multiply through by 7 and introduce a new constant :

Therefore,

In our next example, we will see how to use these rules of indefinite integration together with the power rule of integration.

### Example 5: Integrating Trigonometric Functions Involving Reciprocal Trigonometric Functions

Determine

### Answer

We are asked to evaluate the indefinite integral of the sum or difference of three terms. Letβs start by rewriting this as three separate integrals:

We can evaluate each of these indefinite integrals separately by recalling the following three indefinite integral rules.

For any nonzero real constant , we have

For any real constant , the power rule for integration tells us

With , this tells us

We can use these three rules to integrate each term separately:

Finally, we can combine all of the constants of integration into a new constant, :

Hence,

In our final example, we will combine one of the trigonometric identities with our indefinite integral rules.

### Example 6: Integrating Reciprocal Trigonometric Functions

Determine

### Answer

In this question, we are asked to find the indefinite integral of a trigonometric function. However, in its current form, we do not know how to integrate this function directly. Instead, we can start by rewriting our integrand using the following reciprocal trigonometric identity:

Using this identity and our rules for definite integrals, we get

We can now evaluate this indefinite integral by recalling the following rule. For any real constant ,

By setting , we have

Then, we expand and set :

Hence,

Letβs finish by recapping some of the key points of this explainer.

### Key Points

- By using the fundamental theorem of calculus and the rules for differentiating trigonometric functions, we are able to demonstrate the following rules for finding the indefinite integral of trigonometric functions.
- For any real constant , not equal to zero, and
variable measured in radians,
- ,
- ,
- .