Integral Of Sin(x) / Sin(3x), Looking For Another Solution

May 23, 2025 by ADMIN 59 views

A Novel Approach to Integrating sin(x) / sin(3x)

The integration of trigonometric functions is a fundamental concept in calculus, with numerous applications in physics, engineering, and other fields. One of the most common techniques used to integrate trigonometric functions is substitution, where we rewrite the function in a more manageable form. However, in some cases, this approach may not be the most efficient or elegant solution. In this article, we will explore an alternative method for integrating sin(x) / sin(3x), which provides a unique perspective on this classic problem.

The typical solution to this problem involves rewriting sin(3x) as $3 \sin(x) - 4\sin^3(x)$ , canceling out sin(x), and then multiplying the numerator and denominator by sec^2(x). This results in a logarithmic function with tan(x). While this solution is correct, it may not be the most intuitive or straightforward approach.

In this section, we will present an alternative method for integrating sin(x) / sin(3x). Our approach involves using the identity sin(2x) = 2sin(x)cos(x) to rewrite the function in a more manageable form.

Let's start by rewriting sin(3x) as sin(2x + x). Using the angle addition formula, we can expand this as:

sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x)

Now, we can rewrite sin(x) / sin(3x) as:

sin(x) / (sin(2x)cos(x) + cos(2x)sin(x))

Using the Double Angle Formulas

To simplify this expression, we can use the double angle formulas for sin(2x) and cos(2x). These formulas are:

sin(2x) = 2sin(x)cos(x) cos(2x) = 2cos^2(x) - 1

Substituting these formulas into our expression, we get:

sin(x) / (2sin(x)cos(x)cos(x) + (2cos^2(x) - 1)sin(x))

Simplifying the Expression

Now, we can simplify the expression by combining like terms:

sin(x) / (2sin(x)cos^2(x) + 2cos^2(x)sin(x) - sin(x))

Notice that the first two terms on the right-hand side can be combined as:

2sin(x)cos^2(x) + 2cos^2(x)sin(x) = 2sin(x)cos^2(x) + 2cos^2(x)sin(x)

Using the commutative property of multiplication, we can rewrite this as:

2sin(x)cos^2(x) + 2sin(x)cos^2(x) = 4sin(x)cos^2(x)

Now, we can substitute this back into our expression:

sin(x) / (4sin(x)cos^2(x) - sin(x))

Cancelling Out sin(x)

Notice that sin(x) appears in both the numerator and denominator. We can cancel out sin(x) by dividing both the numerator and denominator by sin(x):

1 / (4cos^2(x) - 1)

Using the Pythagorean Identity

Now, we can use the Pythagorean identity cos^2(x) + sin^2(x) = 1 to rewrite the expression:

1 / (4cos^2(x) - 1) = 1 / (4(1 - sin^2(x)) - 1)

Expanding the denominator, we get:

1 / (4 - 4sin^2(x) - 1) = 1 / (3 - 4sin^2(x))

Using the Substitution u = sin(x)

To simplify this expression further, we can use the substitution u = sin(x). This gives us:

1 / (3 - 4u^2)

Integrating the Expression

Now, we can integrate the expression using the power rule of integration:

∫(1 / (3 - 4u^2)) du

Using the substitution v = 2u, we get:

∫(1 / (3 - 4(1/2)^2v2)) dv

Simplifying the expression, we get:

∫(1 / (3 - v^2)) dv

This is a standard integral, which can be evaluated as:

∫(1 / (3 - v^2)) dv = (1/2)arctan(v/√3) + C

In this article, we presented an alternative method for integrating sin(x) / sin(3x). Our approach involved using the identity sin(2x) = 2sin(x)cos(x) to rewrite the function in a more manageable form. We then used the double angle formulas to simplify the expression and finally integrated the resulting expression using the power rule of integration. This approach provides a unique perspective on this classic problem and highlights the importance of creative problem-solving in mathematics.

[1] "Calculus" by Michael Spivak
[2] "Trigonometry" by I.M. Gelfand
[3] "Integration" by R. Courant

In our previous article, we presented an alternative method for integrating sin(x) / sin(3x). This approach involved using the identity sin(2x) = 2sin(x)cos(x) to rewrite the function in a more manageable form. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the typical solution to this problem?

A: The typical solution to this problem involves rewriting sin(3x) as $3 \sin(x) - 4\sin^3(x)$ , canceling out sin(x), and then multiplying the numerator and denominator by sec^2(x). This results in a logarithmic function with tan(x).

Q: Why is the typical solution not the most intuitive approach?

A: The typical solution involves a lot of algebraic manipulation, which can be confusing and difficult to follow. Our alternative approach, on the other hand, uses a more intuitive and straightforward method to integrate the function.

Q: What is the significance of the identity sin(2x) = 2sin(x)cos(x)?

A: The identity sin(2x) = 2sin(x)cos(x) is a fundamental trigonometric identity that allows us to rewrite the function sin(x) / sin(3x) in a more manageable form. This identity is used extensively in trigonometry and calculus.

Q: How does the substitution u = sin(x) simplify the expression?

A: The substitution u = sin(x) simplifies the expression by allowing us to rewrite the function in terms of a single variable, u. This makes it easier to integrate the function.

Q: What is the final answer to the integral?

A: The final answer to the integral is (1/2)arctan(v/√3) + C, where v = 2u.

Q: Can you provide more examples of how to use this approach?

A: Yes, we can provide more examples of how to use this approach to integrate other trigonometric functions. Please let us know what specific functions you would like to see examples of.

Q: Is this approach applicable to other problems in calculus?

A: Yes, this approach can be applied to other problems in calculus, such as integrating other trigonometric functions or solving differential equations.

Q: What are some common mistakes to avoid when using this approach?

A: Some common mistakes to avoid when using this approach include:

Not using the correct trigonometric identities
Not simplifying the expression correctly
Not using the substitution u = sin(x) correctly
Not integrating the function correctly

In this article, we answered some of the most frequently asked questions about integrating sin(x) / sin(3x). We hope that this article has provided a better understanding of this problem and has helped to clarify any confusion. If you have any further questions or would like to see more examples, please let us know.

[1] "Calculus" by Michael Spivak
[2 "Trigonometry" by I.M. Gelfand
[3] "Integration" by R. Courant