Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.87b

Chapter 8, Problem 8.87b

{Use of Tech} Powers of sine and cosine It can be shown that
∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =
{
[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even
[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd
}
b. Evaluate the integrals with n = 10 and confirm the result.

Verified step by step guidance

Step 1: Recognize the integral formula provided in the problem. The formula states that for n ≥ 2 even, the integral of sinⁿx or cosⁿx from 0 to π/2 is given by [1·3·5···(n-1)]/[2·4·6···n] · π/2. For n ≥ 3 odd, the formula is [2·4·6···(n-1)]/[3·5···n].

Step 2: Identify the value of n in the problem. Here, n = 10, which is an even number. Therefore, we will use the formula for even n: [1·3·5···(n-1)]/[2·4·6···n] · π/2.

Step 3: Compute the numerator of the fraction [1·3·5···(n-1)]. For n = 10, the numerator is the product of all odd numbers from 1 to (n-1), which are 1, 3, 5, 7, and 9.

Step 4: Compute the denominator of the fraction [2·4·6···n]. For n = 10, the denominator is the product of all even numbers from 2 to n, which are 2, 4, 6, 8, and 10.

Step 5: Multiply the fraction obtained in steps 3 and 4 by π/2 to confirm the result of the integral. This will give the final value of the integral ∫ from 0 to π/2 of sin¹⁰x dx or cos¹⁰x dx.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. In this context, the integrals of sine and cosine functions from 0 to π/2 are evaluated, which are crucial for understanding the symmetry and properties of these trigonometric functions over this interval.

Properties of Sine and Cosine Functions

Sine and cosine functions exhibit periodic behavior and are closely related through the identity sin(x) = cos(π/2 - x). This relationship is essential for evaluating integrals involving powers of these functions, as it allows for the interchange of variables and simplification of calculations.

Gamma Function and Factorials

The Gamma function extends the concept of factorials to non-integer values, defined as Γ(n) = (n-1)! for positive integers. In the context of the given integrals, the expressions involving products of odd and even integers can be related to factorials, facilitating the evaluation of the integrals for specific values of n.

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Factorials

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