Problem 61 - Essential Calculus

Question

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta $$

Step-by-Step Solution

Answer

The value of the definite integral \(\int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta\) is computed to be \((-1/2)[\sin(\pi / 4) + (1/5)\sin(5 \cdot \pi / 4)]\).

Step 1: Change the form of the integrand

Use the product-to-sum identity, which states that \( \sin a \sin b = (1/2)[\cos(a - b) - \cos(a + b)] \). Substituting \( a = 2\theta \) and \( b = 3\theta \), we can simplify the integrand to (1/2)[\cos(-\theta) - \cos(5\theta)]. So, the task is to compute the integral \(\int_{0}^{\pi / 4} [(1/2)(\cos(-\theta)- \cos(5\theta))] d\theta \).

Step 2: Split and compute integral

The integral of a difference of two functions is the difference of their integrals. Hence, the task now becomes to compute \((1/2)[\int_{0}^{\pi / 4} \cos(-\theta) d\theta - \int_{0}^{\pi / 4} \cos(5\theta) d\theta]\). The integral of \(\cos(ax)\) is \(\frac{1}{a}\sin(ax)\), using which we find \((1/2)[\sin(-\theta)|_{0}^{\pi / 4} - (1/5)\sin(5\theta)|_{0}^{\pi / 4}]\)

Step 3: Evaluate the limits

Evaluate above expression over the limits \(0\) to \(\pi / 4\). This gives us \((1/2)[\sin(-\pi / 4) - \sin(0) - (1/5)(\sin(5 \cdot \pi / 4) - \sin(0))]\). Simplifying this expression yields the result.

Key Concepts

Product-to-Sum Identities

The product-to-sum identities are useful tools in trigonometry. They help transform products of sines and cosines into sums or differences. This simpler form is often easier to integrate or differentiate. In our exercise, the product-to-sum identity \( \sin a \sin b = \frac{1}{2}[\cos(a - b) - \cos(a + b)] \) is applied. This identity allows us to change the form of the integrand, \( \sin 2\theta \sin 3\theta \), into a form involving cosine functions. By substituting \( a = 2\theta \) and \( b = 3\theta \), we achieve \( \frac{1}{2}[\cos(-\theta) - \cos(5\theta)] \). This transformation simplifies the process of finding the integral, making it more straightforward.

• The key aspect here is recognizing when such transformations are useful and applicable.
• Simplifying expressions can make the integration process easier and more manageable.

Practicing with different identities and transformations builds a solid foundation for solving complex integrals.

Trigonometric Integrals

Trigonometric integrals involve functions such as sine, cosine, tangent, and their combinations. In calculus, integrating these functions is essential for solving problems related to oscillations, waves, and other periodic phenomena. In our example, we used the form \( \frac{1}{2}[\cos(-\theta) - \cos(5\theta)] \). To solve this, the integral is broken down into two separate integrals: \( \int \cos(-\theta) d\theta \) and \( \int \cos(5\theta) d\theta \). The integral of \( \cos(ax) \) is \( \frac{1}{a}\sin(ax) \), where \( a \) is a constant.

• The process often involves using basic trigonometric identities and properties of integrals.
• Simplifying trigonometric expressions beforehand can reduce potential errors when calculating.

Understanding these methods is incredibly useful in both pure and applied mathematics, offering insight into the behavior of physical systems.

Computer Algebra Systems

Computer Algebra Systems (CAS) are software programs designed to perform symbolic mathematics. They can handle a variety of tasks, including differentiation, integration, algebraic simplification, and solving equations. In our exercise, CAS was utilized to calculate the definite integral \( \int_{0}^{\pi / 4} \sin 2\theta \sin 3\theta d\theta \). CAS can be especially beneficial when dealing with complicated integrals or when a quick solution is needed. While understanding the manual steps of solving such problems is important, using CAS can save time and reduce error.

• CAS can swiftly apply transformations and identities, such as product-to-sum, speeding up calculations.
• They provide a visual understanding by allowing students to change parameters and see how results are affected.

Mastering these systems enhances problem-solving skills and prepares students for real-world mathematical challenges, where technology often plays a crucial role.