/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 39

Evaluate the following integrals. $$\int \frac{d x}{\sec x-1}$$

Short Answer

Expert verified
Question: Find the indefinite integral of the following function: $$\int \frac{dx}{\sec{x}-1}$$ Answer: $$\int \frac{dx}{\sec{x}-1} = \ln{|\sin{x}|} + C$$

Step by step solution

01

Rewrite the integrand using trigonometric identities

Recall the trigonometric identity for sec(x): $$\sec{x} = \frac{1}{\cos{x}}$$ Substitute this into the integrand: $$\int \frac{d x}{\sec x-1} = \int \frac{d x}{\frac{1}{\cos x} - 1}$$
02

Simplify the integrand

To simplify the integrand, multiply both the numerator and the denominator by the conjugate of the denominator, which is $$\cos x + 1$$: $$\int \frac{d x}{\frac{1}{\cos x} - 1} = \int \frac{(\cos x + 1) d x}{\left(\frac{1}{\cos x} - 1\right) (\cos x + 1)}$$ Multiply, and we get: $$\int \frac{(\cos x + 1) d x}{1 - \cos^2 x}$$
03

Observe the Pythagorean relation

Notice that we have in the denominator the Pythagorean identity $$1 - \cos^2 x = \sin^2 x$$. So, we can rewrite the integral as: $$\int \frac{(\cos x + 1) d x}{\sin^2 x}$$
04

Use substitution method

Let's make a simple substitution: $$u = \sin{x} \Rightarrow d u = \cos{x} d x$$ Now we have: $$\int \frac{(\cos x + 1) d x}{\sin^2 x} = \int \frac{u d u}{u^2}$$
05

Integrate with respect to u

Now the integration becomes straightforward. Simplify the integral and integrate: $$\int \frac{u d u}{u^2} = \int \frac{d u}{u}$$ Using the elementary integration rule, we have: $$\int \frac{d u}{u} = \ln{|u|} + C = \ln{|\sin{x}|} + C$$ So, the final answer is: $$\int \frac{d x}{\sec x-1} = \ln{|\sin{x}|} + C$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that \(L=\lim _{n \rightarrow \infty}\left(\frac{1}{n} \ln n !-\ln n\right)=-1\) in the following steps. a. Note that \(n !=n(n-1)(n-2) \cdots 1\) and use \(\ln (a b)=\ln a+\ln b\) to show that $$ \begin{aligned} L &=\lim _{n \rightarrow \infty}\left(\left(\frac{1}{n} \sum_{k=1}^{n} \ln k\right)-\ln n\right) \\ &=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \frac{k}{n} \end{aligned} $$ b. Identify the limit of this sum as a Riemann sum for \(\int_{0}^{1} \ln x d x\) Integrate this improper integral by parts and reach the desired conclusion.

A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$

Compare the errors in the Midpoint and Trapezoid Rules with \(n=4,8,16,\) and 32 subintervals when they are applied to the following integrals (with their exact values given). \(\int_{0}^{\pi} \ln (5+3 \cos x) d x=\pi \ln \frac{9}{2}\)

Recall that the substitution \(x=a \sec \theta\) implies either \(x \geq a\) (in which case \(0 \leq \theta<\pi / 2\) and \(\tan \theta \geq 0)\) or \(x \leq-a\) (in which case \(\pi / 2<\theta \leq \pi\) and \(\tan \theta \leq 0\) ). Graph the function \(f(x)=\frac{\sqrt{x^{2}-9}}{x}\) and consider the region bounded by the curve and the \(x\) -axis on \([-6,-3] .\) Then evaluate \(\int_{-6}^{-3} \frac{\sqrt{x^{2}-9}}{x} d x .\) Be sure the result is consistent with the graph.

Sociologists model thespread of rumors using logistic equations. The key assumption is that at any given time, a fraction \(y\) of the population, where \(0 \leq y \leq 1,\) knows the rumor, while the remaining fraction \(1-y\) does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to \(y(1-y) .\) Therefore, the equation that models the spread of the rumor is \(y^{\prime}(t)=k y(1-y)\), where \(k\) is a positive real number. The fraction of people who initially know the rumor is \(y(0)=y_{0},\) where \(0
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.