Problem 34 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \sin 2 x \cos 3 x d x\)

Step-by-Step Solution

Answer

\(-\frac{1}{2} \sin x + \frac{1}{10} \sin 5x + C\)

Step 1: Identify a suitable integral formula

Refer to the table of integrals and look for a formula that resembles the expression \( \sin 2x \cos 3x \). A common formula for the product of sine and cosine is \( \int \sin(ax) \cos(bx) \ dx = \frac{1}{2} \left( \frac{\sin((a-b)x)}{a-b} + \frac{\sin((a+b)x)}{a+b} \right) + C \).

Step 2: Apply the formula

In this case, we have \( a = 2 \) and \( b = 3 \). Substitute these values into the formula: \[\int \sin(2x) \cos(3x) \ dx = \frac{1}{2} \left( \frac{\sin((2-3)x)}{2-3} + \frac{\sin((2+3)x)}{2+3} \right) + C \] This simplifies to: \[\int \sin(2x) \cos(3x) \ dx = \frac{1}{2} \left( \frac{\sin(-x)}{-1} + \frac{\sin(5x)}{5} \right) + C \]

Step 3: Simplify the expression

Simplify each of the terms inside the parentheses: \[\frac{1}{2} \left( -\sin(x) + \frac{\sin(5x)}{5} \right) + C \]Combine the terms to get the final simplified result: \[\frac{1}{2} \left( -\sin(x) + \frac{1}{5}\sin(5x) \right) + C \] This can be written as:\[-\frac{1}{2} \sin x + \frac{1}{10} \sin 5x + C\]

Step 4: Verify the solution

Ensure that each step follows logically with correct arithmetic operations and aligns with integral properties. This finalized expression matches the expected integral properties for \( \int \sin(ax) \cos(bx) \).

Key Concepts

Trigonometric Integrals

When dealing with trigonometric integrals, we're focusing on integrating expressions composed of trigonometric functions like sine, cosine, and tangent. These integrals are significant in calculus as they appear frequently in physical problems involving waves, oscillations, and circular motion.

To handle trigonometric integrals:

• Recognize the standard trigonometric integrals, like \( \int \sin(x) \, dx \) or \( \int \cos(x) \, dx \).
• Use identities to simplify complex trigonometric expressions involving products or powers of sine and cosine.
• Apply integration techniques that suit the situation, such as substitution or partial fraction decomposition.

Trigonometric identities can be especially handy. For instance, the double angle formula \( \sin(2x) = 2\sin(x)\cos(x) \) is often used in these contexts. It's always about transforming the integral into a simpler form you can manage.

Integral Tables

Integral tables are like cheat sheets for calculus students. They contain formulas for common integrals, offering a quick reference to solve problems without needing to derive solutions manually. When you encounter an integral that looks complex, checking an integral table can save you time.

Using an integral table involves:

• Identifying the form of the function you're integrating.
• Matching it with the closest formula in the table.
• Substituting the values from your integral into the formula from the table.

This approach simplifies the process, especially for integrals involving parameters like \( \int \sin(ax) \cos(bx) \, dx \), where specific values for \( a \) and \( b \) matter, just as we saw in the original solution to \( \int \sin(2x) \cos(3x) \, dx \). Remember, correctly matching your integral with the table entry is key for a proper solution.

Product-to-Sum Formulas

Product-to-sum formulas are incredibly useful when integrating products of trigonometric functions, such as \( \sin(ax)\cos(bx) \). These formulas transform products into sums, making the integration more straightforward.

The product-to-sum identities:- \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \)- \( \cos(A)\sin(B) = \frac{1}{2}[\sin(A+B) - \sin(A-B)] \)
Applying these formulas can convert a product form into something simpler to integrate directly. For instance, in our integration of \( \int \sin(2x) \cos(3x) \, dx \), we used the identity that seamlessly transformed the product \( \sin(ax)\cos(bx) \) into a manageable sum of sines, leading us to a straightforward integral solution. By using these formulas, you not only simplify the problem but also reduce errors in calculation.