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 8.5.74a

Capitolo 8, Problema 8.5.74a

Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.

Guida verificata passo dopo passo

Start with the integral: \(\int \sec \theta \, d\theta\).

Multiply the integrand by \(\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}\), which is equivalent to multiplying by 1, so the integral becomes \(\int \sec \theta \cdot \frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).

Rewrite the numerator as \(\sec \theta (\sec \theta + \tan \theta) = \sec^2 \theta + \sec \theta \tan \theta\), so the integral is now \(\int \frac{\sec^2 \theta + \sec \theta \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).

Let \(u = \sec \theta + \tan \theta\). Then compute \(\frac{du}{d\theta}\) by differentiating \(u\) with respect to \(\theta\): \(\frac{du}{d\theta} = \sec \theta \tan \theta + \sec^2 \theta\).

Notice that the numerator of the integrand matches \(\frac{du}{d\theta}\), so rewrite the integral as \(\int \frac{du}{u}\), which can be integrated using the natural logarithm function.

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Integration of Trigonometric Functions

This involves techniques to find antiderivatives of trigonometric expressions. Understanding how to manipulate functions like secant and tangent is essential for simplifying integrals and applying substitution methods effectively.

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Introduction to Trigonometric Functions

Multiplying by a Conjugate Expression

Multiplying the integrand by (sec θ + tan θ) / (sec θ + tan θ) is a strategic algebraic step that simplifies the integral. This technique leverages identities to rewrite the integral in a form that is easier to integrate.

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Multiplying & Dividing Functions

U-Substitution Method

U-substitution is a method for integrating composite functions by substituting a part of the integrand with a new variable u. It simplifies the integral by transforming it into a basic form, often turning complicated trigonometric integrals into standard ones.

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Substitution With an Extra Variable

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