/*! 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 13 Use integration tables to find t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 13

Use integration tables to find the integral. $$ \int e^{x} \arccos e^{x} d x $$

Short Answer

Expert verified
The integral of \(e^x \arccos(e^x)\) with respect to 'x' is \(e^x \arccos(e^x) + e^x \sqrt{1-e^{2x}} - \arcsin(e^x) + C\)

Step by step solution

01

Set up the Integrand with 'u' and 'dv'

We choose 'u' and 'dv' such that 'u' becomes simpler after differentiation and 'dv' is easy to integrate. In this case, let \(u = \arccos e^x\) and \(dv = e^x dx\). Thus, compute \(du = \frac{-e^x dx}{\sqrt{1-e^{2x}}}\) and \(v = e^x\).
02

Apply Integration by Parts Formula

Substituting 'u', 'v', 'du', and 'dv' into integration by parts formula \(\int udv = uv - \int vdu\), we get: \(\int e^x \arccos(e^x) dx = e^x \arccos(e^x) - \int e^x \frac{-e^x dx}{\sqrt{1-e^{2x}}}\). This simplifies to: \(e^x \arccos(e^x) + \int \frac{e^{2x}}{\sqrt{1-e^{2x}}} dx\)
03

Simplify the Integral

The integral in the latest expression doesn't have a standard integral form yet. It can be replaced by a standard integral form by substituting \(u = e^x\) and \(du = e^x dx\), and it simplifies to \(\int \frac{u^2 du} {\sqrt{1-u^{2}}}\). This integral can now be found in the table of standard integrals.
04

Evaluate the Simplified Integral Using the Table

Referring to the integral table, we find that \(\int \frac{u^2 du} {\sqrt{1-u^{2}}} = u \sqrt{1-u^{2}} - \arcsin(u) + C\). Substituting back \(u = e^x\), we get \(e^x \sqrt{1-e^{2x}} - \arcsin(e^x) + C\).
05

Combine All the Parts to Get the Final Answer

Concatenating the previous steps, the solution of the original integral is \(e^x \arccos(e^x) + e^x \sqrt{1-e^{2x}} - \arcsin(e^x) + C\)

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

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} d x, u=\sqrt{1-x} $$

Find the area of the region bounded by the graphs of the equations. $$ y=\cos ^{2} x, \quad y=\sin x \cos x, \quad x=-\pi / 2, \quad x=\pi / 4 $$

Continuous Functions In Exercises 73 and \(74,\) find the value of \(c\) that makes the function continuous at \(x=0\). \(f(x)=\left\\{\begin{array}{ll}\frac{4 x-2 \sin 2 x}{2 x^{3}}, & x \neq 0 \\\ c, & x=0\end{array}\right.\)

Consider the integral \(\int_{0}^{3} \frac{10}{x^{2}-2 x} d x\). To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

Explain why \(\int_{-1}^{1} \frac{1}{x^{3}} d x \neq 0\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.