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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.85
Capitolo 5, Problema 5.3.85

Derivatives of integrals Simplify the following expressions.


d/d𝓍 ∫₀ˣ (√1 + t²) dt (Hint: ∫ˣ₋ₓ (√1 + t²) dt = ∫⁰₋ₓ (√1 + t²) dt + ∫ˣ₋ₓ (√1 + t²) dt ) .

Guida verificata passo dopo passo
1
Step 1: Recognize that the problem involves the Fundamental Theorem of Calculus, which states that if F(x) = ∫ₐˣ f(t) dt, then dF/dx = f(x). This theorem will be key in solving the derivative of the integral.
Step 2: Analyze the given integral ∫₀ˣ (√1 + t²) dt. According to the Fundamental Theorem of Calculus, the derivative of this integral with respect to x is simply the integrand evaluated at the upper limit of integration, which is √(1 + x²).
Step 3: Consider the hint provided: ∫ˣ₋ₓ (√1 + t²) dt = ∫⁰₋ₓ (√1 + t²) dt + ∫ˣ₋ₓ (√1 + t²) dt. This suggests breaking the integral into parts, but for the derivative d/d𝓍 ∫₀ˣ (√1 + t²) dt, the hint is not directly necessary since the Fundamental Theorem of Calculus simplifies the process.
Step 4: Apply the Fundamental Theorem of Calculus directly to the integral ∫₀ˣ (√1 + t²) dt. The derivative with respect to x is simply √(1 + x²), as the lower limit of integration (0) does not contribute to the derivative.
Step 5: Conclude that the derivative of the given integral is √(1 + x²). The hint provided is more relevant for breaking down integrals with different limits, but in this case, the direct application of the theorem suffices.

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed as F(b) - F(a). This theorem also implies that the derivative of an integral function is the integrand evaluated at the upper limit of integration.
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Fundamental Theorem of Calculus Part 1

Differentiation under the Integral Sign

Differentiation under the integral sign allows us to differentiate an integral with respect to a parameter. This technique is useful when the limits of integration or the integrand itself depend on a variable, enabling the evaluation of complex integrals by treating them as functions of that variable.
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Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and is expressed as ∫u dv = uv - ∫v du. This method can simplify the integration of more complex expressions, particularly when one function is easily integrable and the other is easily differentiable.
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