Question

Perform the indicated integrations. $$ \int \sin ^{4}\left(\frac{w}{2}\right) \cos ^{2}\left(\frac{w}{2}\right) d w $$

Short answer

Integrate to get \( \frac{3}{16}w - \frac{1}{2}\sin(w) + \frac{1}{32}\sin(2w) + C \).

Step-by-step solution

Simplify using Trigonometric Identities

We start by using the trigonometric identity: \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \). Thus, we can express \( \sin^4\left(\frac{w}{2}\right) \) as \( \left(\sin^2\left(\frac{w}{2}\right)\right)^2 = \left(\frac{1 - \cos(w)}{2}\right)^2 \).

Simplify the Cosine Term

Express \( \cos^2\left(\frac{w}{2}\right) \) using the identity: \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). Therefore, \( \cos^2\left(\frac{w}{2}\right) = \frac{1 + \cos(w)}{2} \).

Combine the Expressions

Substitute the expressions from Steps 1 and 2 into the integral: \[ \int \left(\frac{1 - \cos(w)}{2}\right)^2 \cdot \left(\frac{1 + \cos(w)}{2}\right) \, dw \]. Simplify the integrand further.

Expand the Expression

Expand the product: \( \left(\frac{1 - \cos(w)}{2}\right)^2 \cdot \frac{1 + \cos(w)}{2} \) using binomial expansion and simplify. This becomes: \[ \frac{1}{8} \left((1 - \cos(w))^2 (1 + \cos(w))\right) \].

Further Simplify and Split the Integral

Simplify the expanded expression: \((1 - \cos(w))^2 (1 + \cos(w)) = (1 - 2\cos(w) + \cos^2(w))(1 + \cos(w))\), which further expands to \(1 + \cos(w) - 2\cos(w) - 2\cos^2(w) + \cos^3(w) + \cos(w) - \cos^3(w) \). Split the integral accordingly.

Integrate the Simplified Expression

Split the integrals: \[ \int \left(\frac{1}{8}(1 - 4\cos(w) + \cos(w)) \right) dw \]. This simplifies to: \( \frac{1}{8} \left( \int 1 \, dw - 4\int \cos(w) \, dw + \int \cos^2(w) \, dw \right) \).

Solve the Integrals

Solve each part: \( \int 1 \, dw = w \), \( \int \cos(w) \, dw = \sin(w) \), and use \( \int \cos^2(w) \, dw = \frac{1}{2}w + \frac{1}{4}\sin(2w) \). Thus, combining these gives us: \[ \frac{1}{8} \left( w - 4\sin(w) + \frac{1}{2}w + \frac{1}{4}\sin(2w) \right) \].

Combine the Results

Combine the constants and expressions properly: \[ \frac{1}{8} \left( \frac{3}{2}w - 4\sin(w) + \frac{1}{4}\sin(2w) \right) + C \]. Simplify to get the final answer.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that allow us to simplify expressions involving trigonometric functions like sine and cosine. In the integration problem, we used identities to transform complex expressions into simpler forms. For instance, the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) helped us express \( \sin^4\left(\frac{w}{2}\right) \) in a simplified form. Similarly, \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) was used to simplify \( \cos^2\left(\frac{w}{2}\right) \).
By applying these identities, the expression under the integral becomes manageable and sets the stage for further algebraic simplification. This process is crucial as it often leads to integrals that are easier to evaluate, eliminating terms or reducing powers. Mastery of these identities will undoubtedly enhance your ability to solve trigonometry-related integration problems effectively.

Definite Integrals

Definite integrals are used to calculate the exact area under a curve, between two points on the x-axis. While our exercise doesn't specifically ask for a definite integral, understanding them is helpful in solving complex mathematical problems involving integration.
Unlike indefinite integrals, which include a constant \( C \), definite integrals compute a specific numerical value. They are evaluated based on a lower and an upper limit: \( \int_{a}^{b} f(x) \, dx \). When evaluating, the Fundamental Theorem of Calculus is often applied. This theorem establishes the link between derivatives and integrals, affirming that integration can reverse the process of differentiation.
When working with definite integrals, after integrating, we replace the variable with the upper limit and subtract the result of replacing it with the lower limit. For example, if \( F(x) \) is the antiderivative of \( f(x) \), then \( \int_{a}^{b} f(x)\, dx = F(b) - F(a) \). This technique is essential in various fields such as physics, engineering, and economics for finding areas, volumes, and solving real-world problems.

Binomial Expansion

The binomial expansion is a powerful algebraic tool used to expand expressions raised to a power, such as \((a + b)^n\). It is particularly useful in the integration process when dealing with complex expressions. In our solution, it was crucial in the step where we expanded the product \((1 - \cos(w))^2 (1 + \cos(w))\).
In a binomial expansion, each term follows the pattern \(( \text{coefficient}) \cdot (\text{first term})^k \cdot (\text{second term})^{n-k} \), where the coefficients are derived from Pascal’s Triangle or can be calculated using combinations: \( \binom{n}{k} \). This step simplifies the expression by breaking it down into individual terms, often making integration straightforward.
Understanding and applying the binomial expansion allows you to tackle a range of problems involving polynomial expressions and is equally important for simplifying complex integrals.

Trigonometric Functions

Trigonometric functions such as sine and cosine are foundational in mathematics, appearing in various applications including problem-solving in calculus. These functions describe the relationships in triangles but extend beyond that in periodic calculations and wave analysis.
In the integration example, knowing the intrinsic properties of \( \sin(w) \) and \( \cos(w) \) was crucial. The integration of such functions follows specific patterns, like \( \int \cos(w) \, dw = \sin(w) + C \), and it’s these patterns that often make solving integrals with trigonometric components possible.
Being proficient with these trigonometric functions allows for a deep understanding of oscillatory behavior, essential for exploring topics in signal processing, electromagnetism, acoustics, and even advanced geometry. Their periodic nature is fundamental for describing repetitions and cycles in these, and various other fields.