Problem 6 - Calculus Early Transcendentals

Question

Use a table of integrals to determine the following indefinite integrals. $$\int \sin 3 x \cos 2 x d x$$

Step-by-Step Solution

Answer

Question: Evaluate the indefinite integral $\int \sin 3x \cos 2x d x$. Solution: Using trigonometric identities, the indefinite integral simplifies to $\int \sin 3x \cos 2x d x = \frac{1}{2} \left[ \frac{-1}{5} \cos(5x) - \cos(x) \right] + C$

Step 1: Simplify the integrand using trigonometric identities

Recall the Product-to-Sum identity for trigonometric functions: $$\sin A \cos B = \frac{1}{2}(\sin(A + B) + \sin(A - B))$$ Using this identity, we can simplify the given integral: $$\int \sin 3x \cos 2x d x = \int \frac{1}{2}(\sin(3x + 2x) + \sin(3x - 2x)) d x$$

Step 2: Perform the integration

Now that the integrand is simplified, we can use the table of integrals to find the antiderivative. In this case, we refer to the integral of sin(ax) and cos(ax), which are respectively: $$\int \sin(ax) d x = \frac{-1}{a} \cos(ax) + C$$ Using this formula, integrate each term inside the integral: $$\int \frac{1}{2}(\sin(5x) + \sin(x)) d x = \frac{1}{2} \left[ \int \sin(5x) d x + \int \sin(x) d x \right]$$

Step 3: Apply the integral formulas and simplify

Apply the integral formulas for sin(ax) and sin(x): $$\frac{1}{2} \left[ \int \sin(5x) d x + \int \sin(x) d x \right] = \frac{1}{2} \left[ \frac{-1}{5} \cos(5x) + \int_{-\frac{1}{1}} \cos(x) d x \right]$$ $$= \frac{1}{2} \left[ \frac{-1}{5} \cos(5x) - \cos(x) \right] + C$$ The final indefinite integral is: $$\int \sin 3 x \cos 2 x d x = \frac{1}{2} \left[ \frac{-1}{5} \cos(5x) - \cos(x) \right] + C$$

Key Concepts

Trigonometric Identity

Trigonometric identities are mathematical tools that express relationships between trigonometric functions. These identities are useful in simplifying expressions and solving equations involving

• sines,
• cosines,
• and other trigonometric functions.

For example, one commonly used identity is the Product-to-Sum identity. This transforms products of trigonometric functions into sums or differences. This specific identity is \[ \sin A \cos B = \frac{1}{2}(\sin(A + B) + \sin(A - B)) \]This equation helps simplify the integrand by expressing a product of two trigonometric functions, such as \(\sin 3x \cos 2x \), as a sum. Once simplified, the expression can be more easily integrated.

Product-to-Sum Identity

The Product-to-Sum identity is a specific trigonometric identity that allows us to convert a product of sine and cosine into a simpler sum of terms. This is particularly beneficial for integration, since sums are generally easier to handle than products.For this exercise, we applied the identity:\[ \sin 3x \cos 2x = \frac{1}{2}(\sin(5x) + \sin(x)) \]Breaking down a product into a sum like this is key when solving integrals involving products of trigonometric functions. It transforms a complex multiplication into an addition, which can often be directly integrated using standard formulas like in our reference table.

Antiderivative

Antiderivatives, often called indefinite integrals, are the opposite of derivatives. They represent the family of functions whose derivative is the given function. In other words, integrating a function gives us its antiderivative.For example, in this exercise:

• We found the antiderivative of \(\sin(5x)\) using the formula: \(\int \sin(ax) \,dx = \frac{-1}{a} \cos(ax) + C \)
• Similarly, for \(\sin(x)\), the antiderivative is \(-\cos(x) + C \)

An important point when finding antiderivatives is to include the constant of integration, \(C\). This constant accounts for any constant that might have been present before differentiation.

Table of Integrals

A table of integrals is a collection of formulas that provide the integral for common functions. These tables are invaluable tools for students and professionals alike since they offer shortcuts for calculating integrals without needing to perform lengthy integrations manually.When dealing with trigonometric functions, tables typically include:

• integrals of \(\sin(ax)\)
• integrals of \(\cos(ax)\)
• and other basic integrals

In this exercise, the table of integrals helped us quickly determine the antiderivatives of \(\sin(5x)\) and \(\sin(x)\). Each corresponding formula was used to simplify our integration process, allowing us to find the indefinite integral of the original trigonometric product with ease.