/*! 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 19 Find the integral. (Note: Solve ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 19

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x \cos x d x $$

Short Answer

Expert verified
The integral of \(x \cos x dx\) is \(x\sin x + \cos x + C\).

Step by step solution

01

Apply the formula for integration by parts

Recall the formula for Integration by Parts: \[\int u dv = u v - \int v du\]For our given integral, we choose:\(u=x\) and \(dv=\cos x dx\).Differentiate and integrate to find \(du\) and \(v\):\(du=dx\) and \(v=\int \cos x dx = \sin x\).Substitute into the formula:\[\int x \cos x dx = x \cdot \sin x - \int \sin x dx\]
02

Compute the remaining integral

The remaining integral can be computed directly. The antiderivative of \(\sin x\) is \(-\cos x\). Using this, we get:\[x \sin x - \int \sin x dx = x \cdot \sin x -(-\cos x) = x\sin x + \cos x\]Thus, the value of the integral \(\int x \cos x dx = x\sin x + \cos x\).
03

Add the constant of integration

Finally, don't forget to add the constant of integration \(C\), to indicate the family of possible solutions. This gives the final answer\[x\sin x + \cos x + C\]

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

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

Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)

Evaluate the definite integral. $$ \int_{0}^{\pi / 4} \tan ^{3} x d x $$

In Exercises 59 and \(60,\) (a) explain why L'H么pital's Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\)

A "semi-infinite" uniform rod occupies the nonnegative \(x\) -axis. The rod has a linear density \(\delta\) which means that a segment of length \(d x\) has a mass of \(\delta d x .\) A particle of mass \(m\) is located at the point \((-a, 0)\). The gravitational force \(F\) that the rod exerts on the mass is given by \(F=\int_{0}^{\infty} \frac{G M \delta}{(a+x)^{2}} d x\) where \(G\) is the gravitational constant. Find \(F\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

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.