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Chapter 8: Problem 54

Evaluate the integrals $$\int \sin 2 x \cos 3 x \, d x$$

Short Answer

Expert verified
The integral evaluates to \(-\frac{1}{10} \cos(5x) + \frac{1}{2} \cos(x) + C\).

Step by step solution

01

Apply Product-to-Sum Identity

The given integral is \( \int \sin 2x \cos 3x \, dx \). Start by applying the product-to-sum identity, which states \( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \). In this case, we have \( A = 2x \) and \( B = 3x \). This gives: \[ \sin 2x \cos 3x = \frac{1}{2}(\sin(5x) + \sin(-x)) \]. Since \( \sin(-x) = -\sin(x) \), the expression simplifies to \[ \frac{1}{2}(\sin(5x) - \sin(x)) \].
02

Rewrite the Integral using the Identity

Substitute the result from Step 1 into the integral: \[ \int \sin 2x \cos 3x \, dx = \int \frac{1}{2}(\sin(5x) - \sin(x)) \, dx \]. Distribute the integral over the sum: \[ = \frac{1}{2} \left( \int \sin(5x) \, dx - \int \sin(x) \, dx \right) \].
03

Evaluate Each Integral

Evaluate the integrals separately. The integral of \( \sin(kx) \) with respect to \( x \) is \( -\frac{1}{k}\cos(kx) + C \), where \( C \) is the constant of integration. So:- \( \int \sin(5x) \, dx = -\frac{1}{5}\cos(5x) + C_1 \).- \( \int \sin(x) \, dx = -\cos(x) + C_2 \).
04

Substitute and Simplify

Substitute the results from Step 3 back into the equation and simplify: \[\frac{1}{2} \left( -\frac{1}{5} \cos(5x) + \cos(x) \right) + C = -\frac{1}{10} \cos(5x) + \frac{1}{2} \cos(x) + C \]. Here, \( C = \frac{1}{2}(C_1 - C_2) \) represents a constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product-to-Sum Identity
The Product-to-Sum Identity is a trigonometric identity that is particularly useful in integration. It allows us to simplify the multiplication of sine and cosine functions into a sum or difference of functions, making the integral easier to solve.
  • For example, the identity \( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \), helps translate the original problem into a more manageable form.
  • When applying this identity to \( \int \sin 2x \cos 3x \, dx \), \( A = 2x \) and \( B = 3x \). Use the identity to transform it into a sum: \( \frac{1}{2}(\sin(5x) - \sin(x)) \).
Recognizing and utilizing these identities simplifies complex expressions, allowing for integration to be performed on simpler functions, significantly easing the computational load.
Trigonometric Integration
Trigonometric integration is a technique used to evaluate integrals that involve trigonometric functions.
  • In our problem, after using the Product-to-Sum Identity, we have separated it into two integrals: \( \int \sin(5x) \, dx \) and \( \int \sin(x) \, dx \).
  • These can be integrated using the standard integral formula for sine, \( \int \sin(kx) \, dx = -\frac{1}{k}\cos(kx) + C \), where \( k \) is a constant and \( C \) is the constant of integration.
Understanding how to apply these trigonometric identities and integrating each part separately makes complicated problems more approachable and solvable.
Constant of Integration
The Constant of Integration is crucial in indefinite integrals, allowing us to account for any constant value that could have been differentiated away.
  • In the expression \( \int f(x) \, dx + C \), the constant \( C \) represents any constant that, when differentiated, results in zero.
  • This is essential because indefinite integrals represent a family of functions rather than a single function.
In our exercise, we found \( C = \frac{1}{2}(C_1 - C_2) \), showcasing how these constants can combine and still represent an arbitrary constant in the solution. Using a constant of integration might seem small, but it is crucial for accurately capturing all potential solutions to an integral.

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