/*! 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 27 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 27

Evaluate the following integrals. $$\int e^{x} \cos x d x$$

Short Answer

Expert verified
Question: Evaluate the integral \(\int e^x \cos x dx\). Answer: \(\int e^x \cos x dx = \frac{1}{2}e^x(\sin x + \cos x) + C\)

Step by step solution

01

Differentiate u to find du

We have \(u = e^x\). Differentiating with respect to x, we get: $$du = e^x dx$$
02

Integrate dv to find v

We have \(dv =\cos x dx\). Integrating with respect to x, we get: $$v = \int \cos x dx = \sin x$$
03

Apply the integration by parts formula

Using the integration by parts formula \(\int u dv = uv - \int v du\), we get: $$\int e^x \cos x dx = e^x \sin x - \int e^x \sin x dx$$ Now, we still have another integral \(\int e^x \sin x dx\) to evaluate. Let's use integration by parts again. This time, let's choose: $$u = e^x$$ $$dv = \sin x dx$$
04

Differentiate u to find du

We have \(u = e^x\). Differentiating with respect to x, we get: $$du = e^x dx$$
05

Integrate dv to find v

We have \(dv = \sin x dx\). Integrating with respect to x, we get: $$v = \int \sin x dx = -\cos x$$
06

Apply the integration by parts formula again

Using the integration by parts formula \(\int u dv = uv - \int v du\), we get: $$\int e^x \sin x dx = -e^x \cos x - \int -e^x \cos x dx$$ Now rewriting our original expression with this new result: $$\int e^x \cos x dx = e^x \sin x - (-e^x \cos x + \int e^x \cos x dx)$$ We still have an integral of \(\int e^x \cos x dx\) in our expression. To solve for it, let's move it to the other side and simplify: $$2\int e^x \cos x dx = e^x \sin x + e^x \cos x$$ Now divide both sides by 2: $$\int e^x \cos x dx = \frac{1}{2}e^x(\sin x + \cos x) + C$$ Finally, we have found the integral: $$\int e^x \cos x dx = \frac{1}{2}e^x(\sin x + \cos x) + C$$

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

By reduction formula 4 in Section 3 $$\int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C$$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(x^{2}-25\right)^{1 / 2},[5,10]$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x-\sqrt[4]{x}} ; x=u^{4}$$

The nucleus of an atom is positively charged because it consists of positively charged protons and uncharged neutrons. To bring a free proton toward a nucleus, a repulsive force \(F(r)=k q Q / r^{2}\) must be overcome, where \(q=1.6 \times 10^{-19} \mathrm{C}\) is the charge on the proton, \(k=9 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}, Q\) is the charge on the nucleus, and \(r\) is the distance between the center of the nucleus and the proton. Find the work required to bring a free proton (assumed to be a point mass) from a large distance \((r \rightarrow \infty)\) to the edge of a nucleus that has a charge \(Q=50 q\) and a radius of \(6 \times 10^{-11} \mathrm{m}.\)

\(\pi < \frac{22}{7}\) One of the earliest approximations to \(\pi\) is \(\frac{22}{7} .\) Verify that \(0 < \int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x=\frac{22}{7}-\pi .\) Why can you conclude that \(\pi < \frac{22}{7} ?\)

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \sqrt{e^{x}+1} d x \text { (Hint: Let } u=\sqrt{e^{x}+1}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.