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 8.5.64

Capitolo 8, Problema 8.5.64

Use any method to evaluate the integrals in Exercises 55–66.
∫ x / (x + √(x² + 2)) dx

Guida verificata passo dopo passo

Start by examining the integral: \(\int \frac{x}{x + \sqrt{x^{2} + 2}} \, dx\). Notice the expression in the denominator involves \(x\) and \(\sqrt{x^{2} + 2}\), which suggests a substitution related to the square root term.

Multiply the numerator and denominator by the conjugate of the denominator to simplify the integrand. The conjugate of \(x + \sqrt{x^{2} + 2}\) is \(\sqrt{x^{2} + 2} - x\). So, multiply numerator and denominator by \(\sqrt{x^{2} + 2} - x\):

\[\frac{x}{x + \sqrt{x^{2} + 2}} \times \frac{\sqrt{x^{2} + 2} - x}{\sqrt{x^{2} + 2} - x} = \frac{x(\sqrt{x^{2} + 2} - x)}{(x + \sqrt{x^{2} + 2})(\sqrt{x^{2} + 2} - x)}.\]

Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^{2} - b^{2}\). Here, \(a = \sqrt{x^{2} + 2}\) and \(b = x\), so the denominator becomes \((\sqrt{x^{2} + 2})^{2} - x^{2} = (x^{2} + 2) - x^{2} = 2\).

Rewrite the integral as \(\int \frac{x \sqrt{x^{2} + 2} - x^{2}}{2} \, dx = \frac{1}{2} \int (x \sqrt{x^{2} + 2} - x^{2}) \, dx\). Now, split the integral into two separate integrals:

\[\frac{1}{2} \int x \sqrt{x^{2} + 2} \, dx - \frac{1}{2} \int x^{2} \, dx.\]

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Integration Techniques

Integration techniques include various methods such as substitution, integration by parts, and rationalizing expressions. Choosing the right technique simplifies the integral and makes it solvable. For this integral, substitution is often effective to handle the square root expression.

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Integration by Parts for Definite Integrals

Substitution Method

The substitution method involves changing variables to simplify the integral. By setting a part of the integrand as a new variable, the integral can be transformed into a more manageable form. This is especially useful when the integrand contains composite functions like square roots.

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Handling Radicals in Integrals

Integrals involving radicals often require algebraic manipulation such as rationalizing the denominator or numerator. Recognizing patterns like expressions under square roots can help in simplifying the integral before applying standard methods.

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