Ch. 8 - Techniques of Integration

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

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.5.74a

Chapter 8, Problem 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.

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Related Practice

Textbook Question

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 3 of (2x - 1) dx

Textbook Question

∫ from -1 to 1 of (t³ + 1) dt

Textbook Question

∫ from 0 to 2 of (t³ + t) dt

Textbook Question

Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about

a. The y-axis.

(See Exercise 57 for a graph.)

Textbook Question

Centroid:

Find the centroid of the region cut from the first quadrant by the curve