Question

Perform the indicated integrations. $$ \int \sin 4 y \cos 5 y d y $$

Short answer

The integral is \( -\frac{1}{18} \cos 9y + \frac{1}{2} \cos y + C \).

Step-by-step solution

Identify the Strategy

Since the integrand involves the product of \( \sin 4y \) and \( \cos 5y \), we can apply the product-to-sum identities. The identity \( \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \) is particularly useful in this case.

Apply the Product-to-Sum Identity

Using the identity from Step 1, we have \( \sin 4y \cos 5y = \frac{1}{2} [\sin(4y + 5y) + \sin(4y - 5y)] = \frac{1}{2} [\sin 9y + \sin(-y)] \). Since \( \sin(-y) = -\sin y \), the expression simplifies to \( \frac{1}{2} [\sin 9y - \sin y] \).

Set Up the New Integral

Substitute the expression from Step 2 into the integral: \( \int \sin 4y \cos 5y \, dy = \int \frac{1}{2} [\sin 9y - \sin y] \, dy \). This can be split into two separate integrals: \( \frac{1}{2} \int \sin 9y \, dy - \frac{1}{2} \int \sin y \, dy \).

Integrate Each Term

First, integrate \( \sin 9y \): The integral of \( \sin(kx) \) is \(-\frac{1}{k} \cos(kx) \). Here \( k = 9 \), so \( \int \sin 9y \, dy = -\frac{1}{9} \cos 9y + C_1 \). For \( \sin y \), the integral is \(-\cos y + C_2 \).

Combine the Results

Substitute the integrals from Step 4 back into the expression: \( \frac{1}{2} \left( -\frac{1}{9} \cos 9y + C_1 \right) - \frac{1}{2} (-\cos y + C_2) \). Simplify this to obtain the final result: \( -\frac{1}{18} \cos 9y + \frac{1}{2} \cos y + C \), where \( C \) is a constant that combines \( C_1 \) and \( C_2 \).

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental in calculus, especially when dealing with integration problems involving trigonometric functions. These are equations that are true for all values of the variables involved. In our example, the integral \( \int \sin 4y \cos 5y \, dy \) involves products of sine and cosine. A basic understanding of identities, such as \( \sin^2 x + \cos^2 x = 1 \), allows for the manipulation and simplification of functions to make integration possible.
\( \sin(A \pm B), \cos(A \pm B) \) can often be rewritten in simpler forms using identities. This broadens the scope of functions we can integrate by traditional means. Knowing and applying these identities quickly transforms complex integrals into manageable ones.

Product-to-Sum Formulas

Product-to-sum formulas are invaluable tools in integrating products of trigonometric functions. These formulas, such as \( \sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \), allow us to express products of sines and cosines as sums, which are simpler to integrate.
In the given exercise, this technique converts the original integrand \( \sin 4y \cos 5y \) into \( \frac{1}{2} [\sin 9y - \sin y] \). This transformation simplifies the integration process by changing a product of functions into a linear combination of simpler functions.
Becoming familiar with these formulas is crucial for efficiently tackling integrations that involve multiple trigonometric terms. They reduce the complexity by transforming the integrals into basic sine and cosine functions.

Antiderivatives

Antiderivatives form the backbone of integration; they provide a function whose derivative matches the given integrand. For trigonometric functions, understanding their antiderivatives is key to solving the integral.For example, the antiderivative of \( \sin(kx) \) is \(-\frac{1}{k}\cos(kx) + C\). In our solution, knowing the antiderivatives allows us to integrate each term separately:

• \( \int \sin 9y \, dy = -\frac{1}{9} \cos 9y + C_1 \)
• \( \int \sin y \, dy = -\cos y + C_2 \)

The constant \( C \) reflects that there are infinite functions differing by a constant that satisfy the derivative theorem. This understanding aids in performing precise integration tasks.

Integration Techniques

In calculus, integration techniques are essential for solving integrals effectively. These techniques are varied, catering to different forms of functions. Familiarity with them aids in simplifying complex integrals. By incorporating techniques such as substitution, by-parts, and trigonometric identities, we tackle the integral efficiently. In our case, using the product-to-sum formula is a technique that simplifies the expression \( \sin 4y \cos 5y \) initially.After applying this transformation, breaking the integral into manageable parts and solving each allows us to determine the integral quickly:

• Split the problem into simple known integrals
• Integrate each part independently
• Combine results and simplify

Such techniques ensure that integrals involving even complex combinations of functions become straightforward to compute.