Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.6.6
Capitolo 8, Problema 8.6.6

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx

Guida verificata passo dopo passo
1
Start by recognizing that the integral is of the form \(\int x (7x + 5)^{3/2} \, dx\), which suggests using substitution to simplify the expression inside the integral.
Let \(u = 7x + 5\). Then, compute the differential \(du = 7 \, dx\), which implies \(dx = \frac{du}{7}\).
Express \(x\) in terms of \(u\) by rearranging the substitution: \(x = \frac{u - 5}{7}\).
Rewrite the integral entirely in terms of \(u\): replace \(x\) with \(\frac{u - 5}{7}\) and \(dx\) with \(\frac{du}{7}\), so the integral becomes \(\int \left( \frac{u - 5}{7} \right) u^{3/2} \cdot \frac{du}{7}\).
Simplify the constants and the integrand, then use the table of integrals to find the integral of the resulting expression in terms of \(u\). After integrating, substitute back \(u = 7x + 5\) to express the answer 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:
5m

Concetti chiave

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

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function, such as (7x + 5)^(3/2).
Video consigliato:
04:27
Substitution With an Extra Variable

Using a Table of Integrals

A table of integrals provides formulas for common integral forms, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form and matching it to an entry in the table can save time and reduce errors. It is important to adjust the integral to fit the table's format, often using substitution.
Video consigliato:
Percorso guidato
08:01
Integration Using Partial Fractions

Power Functions and Exponents in Integration

Integrating functions with fractional exponents, like (7x + 5)^(3/2), requires understanding how to handle power functions. The power rule for integration extends to fractional powers, where the integral of x^n is (x^(n+1))/(n+1) plus a constant, provided n ≠ -1. Recognizing this helps in applying the correct formula or substitution.
Video consigliato:
7:39
Introduction to Exponent Rules
Pratica correlata
Domanda del libro di testo

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

23
views
Domanda del libro di testo

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ z(ln z)² dz

21
views
Domanda del libro di testo

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ √(x - x²) / x dx

22
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ arcsin(y) dy

21
views
Domanda del libro di testo

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (√x / (1 + x³)) dx

Hint: Let u = x^(3/2).

20
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

16
views