Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.4.61a

Capitolo 8, Problema 8.4.61a

Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.

Guida verificata passo dopo passo

Identify the integral to solve: \(\int x^{3} \sqrt{1 - x^{2}} \, dx\).

For integration by parts, choose parts of the integrand as \(u\) and \(dv\). A good choice is to let \(u = x^{2}\) (since \(x^{3} = x \cdot x^{2}\)) and \(dv = x \sqrt{1 - x^{2}} \, dx\) to simplify the integral after differentiation and integration.

Compute \(du\) by differentiating \(u\): \(du = 2x \, dx\).

Find \(v\) by integrating \(dv\): \(v = \int x \sqrt{1 - x^{2}} \, dx\). To do this, use a substitution such as \(t = 1 - x^{2}\), then express \(v\) in terms of \(t\) and integrate.

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), and \(du\) into this formula to rewrite the original integral in terms of simpler integrals to evaluate.

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Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.

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Percorso guidato

06:18

Integration by Parts for Definite Integrals

Substitution Method

Substitution involves changing variables to simplify an integral. By setting a part of the integrand as a new variable, the integral can be rewritten in a simpler form. This is particularly useful when the integrand contains composite functions, such as √(1 - x²), which suggests substituting u = 1 - x².

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07:33

Euler's Method

Handling Powers and Roots in Integrals

Integrals involving powers and roots require careful algebraic manipulation. Expressing roots as fractional exponents and simplifying powers can make integration more straightforward. Recognizing how to rewrite expressions like √(1 - x²) as (1 - x²)^(1/2) helps in applying integration techniques effectively.

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Intro to Power Series

Pratica correlata

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Centroid:

Find the centroid of the region cut from the first quadrant by the curve