/*! 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 Find the following indefinite in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 39

Find the following indefinite integrals. \(\int(\cos x-\sin x) d x\)

Short Answer

Expert verified
\text{sin} x + \text{cos} x + C

Step by step solution

01

Break Down the Integral

Consider the integral \(\text{鈭�(\text{cos} x - \text{sin} x) dx}\). This integral can be broken down into two separate integrals: \(\text{鈭玕text{cos} x \text{ dx}}\) and \(\text{鈭�-\text{sin} x \text{ dx}}\).
02

Integrate \( \text{cos} x \)

The integral of \( \text{cos} x \) with respect to \( x \) is \( \text{sin} x \). Therefore, \( \text{鈭玕text{cos} x \text{ dx} = \text{sin} x} \).
03

Integrate \( -\text{sin} x \)

The integral of \( -\text{sin} x \) with respect to \( x \) is \( -(-\text{cos} x) = \text{cos} x \). Therefore, \( \text{鈭�-\text{sin} x \text{ dx} = \text{cos} x} \).
04

Combine the Results

Combine the two results obtained in Steps 2 and 3: \( \text{鈭�(\text{cos} x - \text{sin} x) dx = \text{sin} x + \text{cos} x + C} \). The constant \( C \) represents 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.

integration techniques
Integration techniques are methods used to find the antiderivative or integral of a function. One common technique is to break down complex integrals into simpler parts, making them easier to solve. For instance, consider the integral \(\text{鈭�(\text{cos} x - \text{sin} x) dx}\). By splitting it into two separate integrals \(\text{鈭玕text{cos} x \text{ dx}}\) and \(\text{鈭�-\text{sin} x \text{ dx}}\), we can handle each function individually. This method, known as the linearity of the integral, is very powerful in simplifying problems that appear difficult at first glance.
trigonometric functions
Trigonometric functions like \(\text{cos} x\) and \(\text{sin} x\) are fundamental in mathematics, especially in calculus. These functions describe relationships between the angles and sides of triangles. Understanding their properties and behavior is essential. Integration involving trigonometric functions often uses their derivatives as a key tool. For example, the integral of \(\text{cos} x\) is \(\text{sin} x\), and the integral of \(-\text{sin} x\) is \(\text{cos} x\). These integrals depend on the knowledge of trigonometric identities and derivatives, making problems involving trig functions both challenging and rewarding.
calculus
Calculus is the branch of mathematics that deals with continuous change. It comprises two main parts: differential calculus and integral calculus. Integral calculus, which includes finding indefinite integrals, is focused on determining the total accumulation of quantities. Indefinite integrals yield families of functions and include a constant of integration, \(C\), since differentiating a constant results in zero. Understanding the principles of calculus, such as limits, derivatives, and integrals, is crucial for solving a wide range of mathematical problems in various fields like physics, engineering, and economics.

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

Describe \(\cot t\) for \(0

Differentiate (with respect to \(t\) or \(x\) ): \(y=2 \cos 3 t\)

Differentiate (with respect to \(t\) or \(x\) ): \(y=4 \sin t\)

Find the following indefinite integrals. \(\int\left(2 \sin 3 x+\frac{\cos 2 x}{2}\right) d x\)

Convert the following to radian measure. \(450^{\circ},-210^{\circ},-90^{\circ}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.