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 8.2.76

Capitolo 8, Problema 8.2.76

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

Guida verificata passo dopo passo

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

Choose parts for integration by parts: let $u = \sqrt{1 - x^{2}}$ and $dv = dx$.

Compute $du$ and $v$: differentiate $u$ to get $du = \frac{d}{dx} \left( (1 - x^{2})^{1/2} \right) dx = \frac{-x}{\sqrt{1 - x^{2}}} dx$, and integrate $dv$ to get $v = x$.

Apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$, so write $\int \sqrt{1 - x^{2}} \, dx = x \sqrt{1 - x^{2}} - \int x \left( \frac{-x}{\sqrt{1 - x^{2}}} \right) dx$.

Simplify the integral inside: $- \int x \left( \frac{-x}{\sqrt{1 - x^{2}}} \right) dx = \int \frac{x^{2}}{\sqrt{1 - x^{2}}} dx$. Then express $x^{2}$ as $1 - (1 - x^{2})$ to rewrite the integral and separate it into simpler parts, leading to the formula $\int \sqrt{1 - x^{2}} \, dx = \frac{1}{2} x \sqrt{1 - x^{2}} + \frac{1}{2} \int \frac{1}{\sqrt{1 - x^{2}}} dx$.

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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 appropriate u and dv is crucial to simplify the integral effectively.

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Integration by Parts for Definite Integrals

Algebraic Manipulation of Integrals

Algebraic manipulation involves rewriting integrals to isolate terms or express them in a more manageable form. In this problem, expressing the integral in terms of itself and another integral helps derive the desired formula, facilitating further evaluation or simplification.

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Integrals Involving Square Roots of Quadratic Expressions

Integrals containing expressions like √(1 - x²) often relate to trigonometric substitutions or geometric interpretations. Recognizing these forms helps in choosing substitution methods or integration techniques, as they commonly appear in problems involving circles or arcs.

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Integrals Involving Natural Logs: Substitution Example 7

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