/*! 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 54 Finding an Indefinite Integral I... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 54

Finding an Indefinite Integral In Exercises \(47-56\) , find the indefinite integral. Use a computer algebra system to confirm your result. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$

Short Answer

Expert verified
The indefinite integral of \( \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} \) is \( \frac{\tan^{2}{x}}{2} - \sin{x} + C \).

Step by step solution

01

Split the Fraction

Break down the complex fraction into two simpler fractions as: \( \int \frac{\sin ^{2} x}{\cos x} dx - \int \frac{\cos ^{2} x}{\cos x} dx \)
02

Simplification

Simplify the fractions to get: \( \int \sin{x} \tan{x} dx - \int \cos{x} dx \)
03

Substitution

Let \( \tan{x} \) be \( u \). Then, \( \sec^{2}{x}dx \) becomes \( du \). So, the first integral becomes \( \int u du \). And, the integral of the second term \( \int \cos{x} dx \) is straightforward.
04

Solve the Integrals

The first integral solves to \( \frac{u^{2}}{2} \) and the second integral solves to \( \sin{x} \). Substitute back \( u = \tan{x} \) to get the final result.
05

Write the Final Integral With Constant of Integration

The final result becomes \( \frac{\tan^{2}{x}}{2} - \sin{x} + C \), where \( C \) is the constant of integration.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Computer Algebra System
A Computer Algebra System (CAS) is a software program designed to perform symbolic mathematics computations efficiently. These systems excel at solving, integrating, and differentiating complex mathematical equations that might be time-consuming by hand. In the context of integrating functions, a CAS can:
  • Save time by swiftly providing accurate results.
  • Handle complex calculations like symbolic integration and simplification.
  • Enable experimentation by allowing users to verify their manual steps.
For the given problem, using CAS helps in confirming the result of the indefinite integral, ensuring that the integration has been performed correctly. This step is crucial in homework or research, where accuracy is paramount.
Trigonometric Substitution
Trigonometric Substitution is a technique used in calculus to simplify integrals involving trigonometric functions or expressions. The idea is to replace a variable with a trigonometric function to make the integral more manageable. In the problem provided, you can identify parts of the expression that can be substituted for simplifying the integration. For example:
  • Recognize that simplifying fractions like \( \frac{\sin^{2}x}{\cos x} \) requires knowledge of trigonometric identities.
  • Substitute \( \tan{x} \) in place of certain expressions to reduce the complexity of the integral.
  • After substitution, rewrite the integral with the new variable for ease of solution.
Understanding when and how to use trigonometric substitution effectively is key in solving integrals with these kinds of expressions.
Integration Techniques
Integration Techniques are a collection of strategies used to solve integrals. Different problems may require different methods, and it is important to identify which technique to use:
  • Substitution: Changing the variable to simplify the problem, as used in the exercise.
  • Integration by Parts: Useful when dealing with products of functions, though not used in every case.
  • Partial Fractions: Breaking down complex fractions into simpler parts, often helpful before integrating.
In the exercise, the substitution replaced \( \tan{x} \) with \( u \) and manipulated the expression to simplify both integrals. Each technique requires practice, but mastering them will make solving indefinite integrals more straightforward.
Constant of Integration
The Constant of Integration, denoted usually as \( C \), is an essential aspect of indefinite integrals. When you integrate a function, the result is not just a single function but a family of functions, all differing by a constant. This is because:
  • Integration reverses differentiation, which loses constant terms.
  • The constant \( C \) allows the integration result to encompass all possible antiderivatives.
  • In practical problems, the value of \( C \) might be determined by initial conditions or boundary values.
For the exercise at hand, including the constant of integration, \( C \), in the final expression \( \frac{\tan^{2}{x}}{2} - \sin{x} + C \) ensures it correctly represents all possible solutions to the integral.

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

Normal Probability The mean height of American men between 20 and 29 years old is 70 inches, and the standard deviation is 2.85 inches. A 20 - to 29 -year- old man is chosen at random from the population. The probability that he is 6 feet tall or taller is $$ P(72 \leq x<\infty)=\int_{72}^{\infty} \frac{1}{2.85 \sqrt{2 \pi}} e^{-(x-70)^{2 / 6.245}} d x $$ (a) Use a graphing utility to graph the integrand. Use the graphing utility to convince yourself that the area between the \(x\) -axis and the integrand is \(1 .\) (b) Use a graphing utility to approximate \(P(72 \leq x<\infty)\) . (c) Approximate \(0.5-P(70 \leq x \leq 72)\) using a graphing utility. Use the graph in part (a) to explain why this result is the same as the answer in part (b).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t .\) The Laplace Transform of \(f(t)\) is defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ when the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises \(95-102,\) find the Laplace Transform of the function. $$ f(t)=t $$

Finding Values In Exercises 49 and \(50,\) determine all values of \(p\) for which the improper integral converges. $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$

Comparing Functions In Exercises \(69-74,\) use \(L^{\prime}\) Hopital's Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, g(x)=e^{n x},\) and \(h(x)=(\ln x)^{n},\) where \(n>0, m>0,\) and \(x \rightarrow \infty\) . $$ \lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{3}} $$

Velocity in a Resisting Medium The velocity \(v\) of an object falling through a resisting medium such as air or water is given by $$v=\frac{32}{k}\left(1-e^{-k t}+\frac{v_{0} k e^{-k t}}{32}\right)$$ where \(v_{0}\) is the initial velocity, \(t\) is the time in seconds, and \(k\) is the resistance constant of the medium. Use L'Hopital's Rule to find the formula for the velocity of a falling body in a vacuum by fixing \(v_{0}\) and \(t\) and letting \(k\) approach zero. (Assume that the downward direction is positive.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

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.