Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

5:22

Factorials

Related Practice

Textbook Question

Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).

b. ∫ 7x e³ˣ dx

Textbook Question

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

b. Is L concave up or concave down on [0, ∞)?

views

Textbook Question

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

b. Which car travels farthest on the interval 0 ≤ t ≤ 5?

views

Textbook Question

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.

views

Textbook Question

Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).

b. Find the volume of the solid generated when R is revolved about the x-axis.

views

Textbook Question

68. Different methods

b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.