/*! 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 70 Compute the indefinite integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 70

Compute the indefinite integrals. $$ \int \frac{\cos x}{1-\cos ^{2} x} d x $$

Short Answer

Expert verified
The indefinite integral is \(-\csc x + C\).

Step by step solution

01

Simplify the Integrand

First, simplify the integrand \( \frac{\cos x}{1-\cos^2 x} \). Notice that \( 1 - \cos^2 x = \sin^2 x \) due to the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). So the integrand becomes \( \frac{\cos x}{\sin^2 x} = \cot x \csc x \).
02

Use Trigonometric Identity for Integration

Recognize that \( \int \cot x \csc x \, dx \) is a standard integral. The derivative of \( \csc x \) is \( -\cot x \csc x \). Thus, the integral \( \int \cot x \csc x \, dx = -\csc x + C \).
03

Write the Final Result

Combine the results from the previous step to conclude the indefinite integral. The indefinite integral is given by \( \int \frac{\cos x}{1-\cos^2 x} \, dx = - \csc x + C \), 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.

Trigonometric Identities
Trigonometric identities are fundamental tools in calculus and trigonometry. They help simplify complex expressions into more manageable forms. In this exercise, one such identity becomes particularly useful: the Pythagorean identity. Here's a simple way to think about it:
  • Trigonometric functions, like sine and cosine, are related through various identities that can transform one function into another.
  • These identities come in handy, especially when dealing with integrals, as they help to reduce complex expressions into ones involving simpler or more standard integrals.
  • In this problem, the identity converts the expression under the integral from a fraction involving cosine into a product involving cotangent and cosecant, two other trigonometric functions.
Recognizing which trigonometric identity to use can drastically simplify the process of solving integrals and finding antiderivatives.
Pythagorean Identity
The Pythagorean identity is a cornerstone of trigonometry and is essential for solving integrals involving trigonometric functions. It states that:\[ \sin^2 x + \cos^2 x = 1. \]Using this identity, we can express other trigonometric functions in terms of each other. In this exercise:
  • The expression \( 1 - \cos^2 x \) becomes \( \sin^2 x \) by rearranging the Pythagorean identity.
  • This substitution simplifies the original integrand \( \frac{\cos x}{1 - \cos^2 x} \) into \( \frac{\cos x}{\sin^2 x} \), or \( \cot x \csc x \).
  • This simplification is crucial because integrating the resulting function is more straightforward than integrating the original one.
Mastering the Pythagorean identity allows you to transform and simplify trigonometric expressions effectively, making complex integrations more manageable.
Standard Integral
A standard integral involves familiar, easily integrable functions. Recognizing these can save time and reduce complexity.
  • In this solution, the integrand \( \cot x \csc x \) falls into this category of standard integrals.
  • The derivative of \( \csc x \) is already known to be \( -\cot x \csc x \). Recognizing this pattern, we immediately know that:\[ \int \cot x \csc x \, dx = -\csc x + C. \]
  • This property makes integrating quicker. Instead of performing long algebraic steps, you can rely on familiar results.
By knowing some common standard integrals, such as this one, you can streamline the integration process significantly, especially when trigonometric functions are involved.

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

Compute the indefinite integrals. $$ \int 2^{x} d x $$

Evaluate the definite integrals. $$ \int_{4}^{9} \frac{1+\sqrt{x}}{\sqrt{x}} d x $$

Evaluate the definite integrals. $$ \int_{-2}^{-1} \frac{1}{1-u} d u $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2 x^{2}}^{6}(1+\tan t) d t $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2 x}^{3}(1+\sin t) d t $$
See all solutions

Recommended explanations on Biology Textbooks

Responding to Change

Read Explanation

Ecosystems

Read Explanation

Cell Cycle

Read Explanation

Biological Organisms

Read Explanation

Control of Gene Expression

Read Explanation

Heredity

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.