Domanda del libro di testo
Evaluate the integrals in Exercises 91–102.
96. ∫dy/((arcsin y)(1-y²))
21
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.4.3
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Guida verificata passo dopo passo1
Identify the given function: \(y = \frac{1}{x} \int_{1}^{x} \frac{e^{t}}{t} \, dt\).
Apply the product rule and the Fundamental Theorem of Calculus to find \(y'\). Since \(y\) is a quotient involving an integral, write \(y\) as \(y = \frac{1}{x} \cdot F(x)\) where \(F(x) = \int_{1}^{x} \frac{e^{t}}{t} \, dt\).
Differentiate \(y\) using the quotient rule or product rule: \(y' = \frac{d}{dx} \left( \frac{1}{x} F(x) \right) = -\frac{1}{x^{2}} F(x) + \frac{1}{x} F'(x)\).
Use the Fundamental Theorem of Calculus to find \(F'(x) = \frac{e^{x}}{x}\), then substitute back into the expression for \(y'\).
Substitute \(y\) and \(y'\) into the differential equation \(x^{2} y' + x y\) and simplify to verify that it equals \(e^{x}\).
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:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is the integrand evaluated at that limit. It allows us to differentiate functions defined by integrals, which is essential for finding y' in the given problem.
Video consigliato:
Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1
Differentiation of Functions Defined by Integrals
When a function is given as an integral with variable limits, its derivative involves applying the Fundamental Theorem of Calculus and the product or chain rule if multiplied by other functions. Understanding this helps compute y' for y = (1/x) ∫(1 to x) (e^t / t) dt.
Video consigliato:
Percorso guidato
04:22Integrals Resulting in Basic Trig Functions Example 1
Verification of Solutions to Differential Equations
To verify a function solves a differential equation, substitute the function and its derivative into the equation and simplify. If both sides match, the function is a solution. This process confirms that y and y' satisfy x²y' + xy = e^x.
Video consigliato:
04:00Solutions to Basic Differential Equations
Pratica correlata
Domanda del libro di testo
22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.
37
views
Domanda del libro di testo
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x⁵
16
views
Domanda del libro di testo
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
119. y = (sin x)ˣ
31
views
Domanda del libro di testo
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
15. lim(x→∞)arctan(x)
28
views
Domanda del libro di testo
L’Hôpital’s Rule
Find the limits in Exercises 103–110.
108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)
24
views