Trigonometric Integrals
In this topic, we will study how to integrate certain combinations involving products and powers of trigonometric functions.
We consider
1. Integrals of the form
To evaluate integrals of products of sine and cosine with different arguments, we apply the identities
2. Integrals of the form
We assume here that the powers
To find an integral of this form, use the following substitutions:
- If
(the power of sine) is odd, we use the substitutionand the identity
to express the remaining even power of sine in
terms. - If
(the power of cosine) is odd, we use the substitutionand the identity
to express the remaining even power of cosine in
terms. - If both powers
and are even, we reduce the powers using the half-angle formulas
The integrals of type
3. Integrals of the form
The power of the integrand can be reduced using the trigonometric identity
and the reduction formula
4. Integrals of the form
The power of the integrand can be reduced using the trigonometric identity
and the reduction formula
5. Integrals of the form
This type of integrals can be simplified with help of the reduction formula:
6. Integrals of the form
Similarly to the previous examples, this type of integrals can be simplified by the formula
7. Integrals of the form
- If the power of the secant
is even, then using the identitythe secant function is expressed as the tangent function. The factor
is separated and used for transformation of the differential. As a result, the entire integral (including differential) is expressed in terms of the function - If both the powers
and are odd, then the factor which is necessary to transform the differential, is separated. Then the entire integral is expressed in terms of - If the power of the secant
is odd, and the power of the tangent is even, then the tangent is expressed in terms of the secant using the identityAfter this substitution, you can calculate the integrals of the secant.
8. Integrals of the form
- If the power of the cosecant
is even, then using the identitythe cosecant function is expressed as the cotangent function. The factor
is separated and used for transformation of the differential. As a result, the integrand and differential are expressed in terms of - If both the powers
and are odd, then the factor which is necessary to transform the differential, is separated. Then the integral is expressed in terms of - If the power of the cosecant
is odd, and the power of the cotangent is even, then the cotangent is expressed in terms of the cosecant using the identityAfter this substitution, you can find the integrals of the cosecant.
Solved Problems
Click or tap a problem to see the solution.
Example 1
Calculate the integral
Example 2
Evaluate the integral
Example 3
Find the integral
Example 4
Find the integral
Example 5
Calculate the integral
Example 6
Evaluate the integral
Example 7
Evaluate the integral
Example 8
Evaluate the integral
Example 1.
Calculate the integral
Solution.
Let
Example 2.
Evaluate the integral
Solution.
Making the substitution
we obtain:
Example 3.
Find the integral
Solution.
Using the identities
we can write:
Calculate the integrals in the latter expression.
To find the integral
Then
Hence, the initial integral is
Example 4.
Find the integral
Solution.
The power of cosine is odd, so we make the substitution
We rewrite the integral in terms of
Example 5.
Calculate the integral
Solution.
We can write:
We convert the integrand using the identities
This yields
Example 6.
Evaluate the integral
Solution.
As the power of sine is odd, we use the substitution
The integral is written as
By the Pythagorean identity,
Hence
Example 7.
Evaluate the integral
Solution.
We see that both powers are odd, so we can substitute either
The integral takes the form
Using the Pythagorean identity
we can write
Example 8.
Evaluate the integral
Solution.
The powers of both sine and cosine are odd. Hence we can use the substitution
By the Pythagorean identity,
so we obtain