Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.2.76

Chapter 8, Problem 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.

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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