Problem 83 - Calculus Early Transcendentals

Question

Powers of sine and cosine It can be shown that \(\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x=\) \(\left\\{\begin{array}{ll}\frac{1 \cdot 3 \cdot 5 \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2} & \text { if } n \geq 2 \text { is an even integer } \\ \frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n} & \text { if } n \geq 3 \text { is an odd integer. }\end{array}\right.\)

Step-by-Step Solution

Answer

Based on the given solution, summarize the method to evaluate the integral of sine to the power of n and cosine to the power of n from 0 to π/2 for both even and odd integers. To evaluate the integral of sine to the power of n and cosine to the power of n from 0 to π/2: 1. If n is an even integer: a. Multiply the odd integers less than n. b. Multiply the even integers less than or equal to n. c. Divide the product of odd integers by the product of even integers. d. Multiply the result by π/2. 2. If n is an odd integer: a. Multiply the even integers less than n. b. Multiply the odd integers greater than 1 and less than or equal to n. c. Divide the product of even integers by the product of odd integers.

Step 1: Calculate integral of sin^n(x) and cos^n(x), when n is an even integer

For even integer values of n, the formula to determine the integral of sine to the power of n and cosine to the power of n from 0 to π/2 is given as: \(\int_{0}^{\pi/2} \sin^n x dx = \int_{0}^{\pi/2} \cos^n x dx = \frac{1 \cdot 3 \cdot 5 \cdots (n - 1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2}\) Now, let's break down this formula: First, multiply the odd integers less than n. Second, multiply the even integers less than or equal to n. Next, divide the product of odd integers by the product of even integers. Finally, multiply the result by π/2.

Step 2: Calculate integral of sin^n(x) and cos^n(x), when n is an odd integer

For odd integer values of n, the formula to determine the integral of sine to the power of n and cosine to the power of n from 0 to π/2 is given as: \(\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x=\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n}\) Now, let's break down this formula: First, multiply the even integers less than n. Second, multiply the odd integers greater than 1 and less than or equal to n. Next, divide the product of even integers by the product of odd integers. After following the steps outlined above, we can evaluate the integral of sine to the power of n and cosine to the power of n from 0 to π/2 depending on whether n is an even integer or an odd integer.

Key Concepts

Integration Techniques

Understanding various integration techniques is essential to solve complex calculus problems. Integrals are math operations that, essentially, find the area under a curve. There are several techniques to perform integration:

• Substitution: When a function is too complicated, you can simplify it by substituting a part of the function with a new variable.
• Integration by parts: Useful when integrating the product of two functions, this technique is based on the product rule for differentiation.
• Partial fractions: Used when integrating rational functions, by breaking them down into simpler fractions.
• Trigonometric integrals: These involve the integration of trigonometric functions, often requiring special techniques or identities to solve.

Each method is a tool that, when used appropriately, can simplify the integration process and lead to the solution. When approaching trigonometric integrals, recognize patterns and apply identities or substitution to simplify the expression before integrating.

Trigonometric Integrals

Dealing with trigonometric integrals involves integration of powers and products of sine and cosine functions. When integrating powers of sine and cosine, it's crucial to understand whether the power is even or odd, as this influences the technique used to solve the integral.

Techniques for Powers of Sine and Cosine Integrals

• For even powers, you can use power-reducing formulas to express the powers of sine and cosine in terms of the first power.
• For odd powers, applying the identities and using substitution can turn the integral into a more manageable form.

Often, simplifying the original function using trigonometric identities like the half-angle or double angle formulas can pave the way to use standard integration techniques.

Definite Integrals

Definite integrals are used to calculate the exact area under a curve between two specified points. They differ from indefinite integrals by having upper and lower limits.

Properties of Definite Integrals

• Interval limits: The lower and upper limits define the range of integration.
• Integrands: This is the function being integrated within the given interval.
• Even and Odd functions: If the function is even, its integral over the symmetric interval around the origin is twice the integral from zero to the positive limit. For odd functions, the integral over the symmetric interval is always zero.

In our exercise, the limits are from 0 to \(\frac{\pi}{2}\), and the symmetry in those limits can be applied to simplify the trigonometric integrals, especially considering the properties of even and odd functions.

Odd and Even Functions

Understanding odd and even functions plays a significant role in simplifying integrals. A function f(x) is called even if \(f(x) = f(-x)\) and odd if \(f(-x) = -f(x)\).

Impact on Integration

• Even functions: Have symmetry about the y-axis and integrating these from \( -a \) to \( a \) leaves out duplicate areas, thus \(\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\) when f(x) is even.
• Odd functions: Have symmetry with respect to the origin, which means the areas above and below the x-axis cancel each other out, hence \(\int_{-a}^{a} f(x) dx = 0\) when f(x) is odd.

In the exercise, using the theory behind even and odd integers helps to derive a general formula for the integrals of \(\sin^n x\) and \(\cos^n x\), significantly simplifying the integration process for these particular trigonometric functions.