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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.4.46
Capitolo 8, Problema 8.4.46

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)

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1
Start by examining the integral \( \int \frac{\sqrt{x}}{1 - x^{3}} \, dx \). Notice the hint suggests the substitution \( u = x^{3/2} \). Since \( x^{3/2} = (x^{1/2})^{3} \), this substitution will help simplify the expression inside the denominator.
Express \( u = x^{3/2} \) and differentiate both sides with respect to \( x \) to find \( du \) in terms of \( dx \). Recall that \( \frac{d}{dx} x^{3/2} = \frac{3}{2} x^{1/2} \), so \( du = \frac{3}{2} x^{1/2} dx \).
Solve for \( x^{1/2} dx \) from the expression for \( du \): \( x^{1/2} dx = \frac{2}{3} du \). This substitution will allow you to rewrite the integral in terms of \( u \).
Rewrite the integral in terms of \( u \) using the substitutions: \( \sqrt{x} dx = x^{1/2} dx = \frac{2}{3} du \) and \( 1 - x^{3} = 1 - (x^{3/2})^{2} = 1 - u^{2} \). The integral becomes \( \int \frac{\frac{2}{3} du}{1 - u^{2}} \).
Recognize that the integral now has the form \( \int \frac{du}{1 - u^{2}} \), which is suitable for a trigonometric substitution. Use the substitution \( u = \sin \theta \) or \( u = \cos \theta \) to simplify the denominator and proceed with the integration.

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Substitution Method

The substitution method simplifies integrals by changing variables to make the integral easier to evaluate. By letting u equal a function of x, the integral can be rewritten in terms of u, often reducing complexity. This technique is especially useful when the integral contains composite functions.
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Euler's Method

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving roots or quadratic forms with trigonometric functions to simplify integration. It leverages identities like sin²θ + cos²θ = 1 to transform the integral into a trigonometric form that is easier to solve.
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Introduction to Trigonometric Functions

Handling Integrals with Roots and Rational Expressions

Integrals involving roots and rational expressions often require careful manipulation, such as substitution or trigonometric substitution, to simplify the integrand. Recognizing patterns like √x or expressions like 1 - x³ helps in choosing the right substitution to transform the integral into a solvable form.
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Introduction to Integration by Parts Example 2
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