Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.3.13

Capitolo 8, Problema 8.3.13

9–61. Trigonometric integrals Evaluate the following integrals.
13. ∫ sin⁵x dx

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves an odd power of sine, which suggests using the identity

sin²x = 1 - cos²x

to simplify the expression. Split

sin⁵x

sin⁴x · sinx

Step 2: Rewrite

sin⁴x

using the identity

sin²x = 1 - cos²x

. This gives

sin⁴x = (sin²x)² = (1 - cos²x)²

Step 3: Substitute

sin⁴x

into the integral, resulting in

∫ (1 - cos²x)² · sinx dx

. This prepares the integral for a substitution.

Step 4: Use the substitution

u = cosx

, which implies

du = -sinx dx

. Replace

sinx dx

with

-du

and rewrite the integral in terms of

u

Step 5: Expand

(1 - u²)²

to get

1 - 2u² + u⁴

, and integrate term by term. After integration, substitute back

u = cosx

to express the result in terms of

x

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, particularly in integration. The sine function, sin(x), oscillates between -1 and 1 and is periodic with a period of 2π. Understanding the properties of these functions is essential for evaluating integrals involving them, as they often require specific techniques or identities.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Integration Techniques

Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, or using trigonometric identities are commonly employed. In the case of ∫ sin⁵x dx, recognizing patterns and applying the appropriate technique is crucial for simplifying the integral.

Video consigliato:

Percorso guidato

06:18

Integration by Parts for Definite Integrals

Power Reduction Formulas

Power reduction formulas are used to express higher powers of trigonometric functions in terms of lower powers. For example, the formula for sin²x can be expressed as (1 - cos(2x))/2. This is particularly useful for integrals like ∫ sin⁵x dx, as it allows the integral to be rewritten in a more manageable form, facilitating easier evaluation.

Video consigliato: