/*! 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 25 Find the integral. $$ \int e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 25

Find the integral. $$ \int e^{x} \sqrt{1-e^{2 x}} d x $$

Short Answer

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

Step by step solution

01

Substitute

Set the substitution \(u = e^x\). The differential of \(u\), \(du\), is equal to \(e^x dx\). Hence, the integral can be rewritten as \( \int \sqrt{1 - u^2} du \).
02

Apply Trigonometric Substitution

In situations where the integral involves the square root of \(1 - x^2\), a trigonometric substitution of the form \(x = \sin(\theta)\) can be useful. Replacing \(u\) with \(\sin(\theta)\), the integral becomes \(\int \sqrt{1 - \sin^2(\theta)} d\theta = \int \cos(\theta) d\theta\).
03

Simple Integration

Now, the integral of cos(\theta) is sin(\theta). However, note that in step 2, we made the substitution \(u = \sin(\theta)\). Thus, when integrating \(\cos(\theta)\), we end up with \(u\).
04

Convert Back to Original Variable

Finally, since our original variable was \(x\), not \(u\) or \(\theta\), we have to substitute back in to obtain the answer in terms of \(x\). Remembering that \(u = e^x\), our final answer is \(e^x + C\), 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!

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

For the region bounded by the graphs of the equations, find: (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. $$ y=\cos x, y=0, x=0, x=\pi / 2 $$

Evaluate the definite integral. $$ \int_{0}^{\pi / 2} \frac{\cos t}{1+\sin t} d t $$

(a) Let \(f^{\prime}(x)\) be continuous. Show that \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)\) (b) Explain the result of part (a) graphically.

In L'H么pital's 1696 calculus textbook, he illustrated his rule using the limit of the function \(f(x)=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}\) as \(x\) approaches \(a, a>0 .\) Find this limit.

Evaluate the definite integral. $$ \int_{-\pi}^{\pi} \sin 3 \theta \cos \theta d \theta $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure 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.