Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - x²)^(1/2) / x⁴ dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.4
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (1 / (cos² x tan x)) dx from π/3 to π/4
Verified step by step guidance1
Start by rewriting the integrand \(\frac{1}{\cos^{2} x \tan x}\) in terms of sine and cosine functions to simplify the expression. Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute this into the integrand.
After substitution, simplify the integrand: \(\frac{1}{\cos^{2} x \cdot \frac{\sin x}{\cos x}} = \frac{1}{\cos^{2} x} \cdot \frac{\cos x}{\sin x} = \frac{\cos x}{\cos^{2} x \sin x} = \frac{1}{\cos x \sin x}\).
Recognize that the integrand is now \(\frac{1}{\sin x \cos x}\). Use the identity \(\sin 2x = 2 \sin x \cos x\) to rewrite the integrand as \(\frac{1}{\sin x \cos x} = \frac{2}{\sin 2x}\).
Rewrite the integral with the new integrand: \(\int_{\pi/3}^{\pi/4} \frac{2}{\sin 2x} \, dx\). Consider a substitution to simplify the integral further, such as \(u = 2x\), which implies \(du = 2 dx\) or \(dx = \frac{du}{2}\).
Change the limits of integration accordingly: when \(x = \pi/3\), \(u = 2 \cdot \pi/3 = \frac{2\pi}{3}\); when \(x = \pi/4\), \(u = 2 \cdot \pi/4 = \frac{\pi}{2}\). Substitute into the integral and simplify before proceeding to integrate \(\csc u\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify integrals by rewriting expressions in more manageable forms, such as converting tan x and cos² x into sine and cosine functions. Using identities like tan x = sin x / cos x helps transform the integral into a solvable form.
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Substitution Method
The substitution method involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can be rewritten in terms of this variable, making it easier to evaluate. This technique is especially useful when the integral contains composite functions or products of functions and their derivatives.
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Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two specific points, called limits of integration. Evaluating a definite integral requires finding the antiderivative and then applying the Fundamental Theorem of Calculus by substituting the upper and lower limits. Understanding how to handle these limits is essential for obtaining the final numerical value.
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