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 8.9.4

Capitolo 8, Problema 8.9.4

4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.

Guida verificata passo dopo passo

Rewrite the integral \( \int_0^1 \frac{1}{x^{1/5}} \, dx \) as \( \int_0^1 x^{-1/5} \, dx \) to express the integrand with a negative exponent, which is easier to integrate.

Recognize that the integrand \( x^{-1/5} \) is undefined at \( x = 0 \) because it involves division by zero, so the integral is an improper integral and must be expressed as a limit.

Express the integral as a limit approaching 0 from the right: \[ \lim_{t \to 0^+} \int_t^1 x^{-1/5} \, dx \]. This handles the improper behavior at the lower limit.

Find the antiderivative of \( x^{-1/5} \) using the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \( n = -\frac{1}{5} \). So, \[ \int x^{-1/5} \, dx = \frac{x^{4/5}}{4/5} + C = \frac{5}{4} x^{4/5} + C \].

Evaluate the definite integral by substituting the limits into the antiderivative and then take the limit as \( t \to 0^+ \): \[ \lim_{t \to 0^+} \left( \frac{5}{4} (1)^{4/5} - \frac{5}{4} t^{4/5} \right) \].

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.

Improper Integrals and Limits

An integral is improper if the integrand is unbounded or the interval is infinite. To evaluate such integrals, rewrite them as limits approaching the problematic point, ensuring the integral converges or diverges.

Video consigliato:

Percorso guidato

11:11

Improper Integrals: Infinite Intervals

Power Rule for Integration

The power rule states that ∫ x^n dx = (x^(n+1)) / (n+1) + C for n ≠ -1. This rule helps integrate functions with variable exponents, such as x^(-1/5), by increasing the exponent by one and dividing by the new exponent.

Video consigliato:

Percorso guidato

04:04

Power Rule for Indefinite Integrals

Evaluating Definite Integrals

Definite integrals compute the net area under a curve between two limits. After finding the antiderivative, substitute the upper and lower limits and subtract to find the integral's value.

Video consigliato:

Percorso guidato

05:43

Definition of the Definite Integral

Pratica correlata

Domanda del libro di testo

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

15. x / ((x⁴ - 16)²)

views

Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

37. ∫ dx / √(x² + 10x), x >

117

views

Domanda del libro di testo

11. ∫ 3u / (2u + 7) du

views

Domanda del libro di testo

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.

f(x) = 1/(x√(x² - 36)), [12/√3 , 12]

views