/*! 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 29 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 29

Find the indefinite integral. $$ \int e^{x} \sin e^{x} d x $$

Short Answer

Expert verified
The indefinite integral is \(I = \frac{1}{2}e^x \sin e^x + C\)

Step by step solution

01

Setting Up the Integral

The first step will involve applying integration by parts to the original function. Using the formula \(\int u v dx= u \int v dx - \int u' (\int v dx) dx\), let \(u=e^x\) and \(v=\sin(e^x)\). The differentials of these will be \(du=e^xdx\) and \(dv=\cos(e^x)e^xdx\). But in this case choosing \(u\) and \(v\) doesn't make the integral simpler. So, in order to simplify the integral we will need to somewhat guess the form of the solution, and make an assumption that the integral can be written in form \(I = e^x \sin e^x - I\), where \(I\) represents the original integral. Now, solving for \(I\) from this equation will help proceed to the final steps.
02

Solving for Integral I

To solve for \(I\), first add \(I\) to both sides of the equation to get \(2I = e^x \sin e^x\). Then divide by \(2\) to solve for \(I\), yielding \(I = \frac{1}{2}e^x \sin e^x\).
03

Finding the Final Solution

Include the constant of integration to adjust for any constant in original function that disappeared when differentiating to get the original function. The final expression for \(I\) can be written as \(I= \frac{1}{2}e^x \sin e^x + C\), where \(C\) is the constant of integration.

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Most popular questions from this chapter

solve the equation for \(\theta\) \((0 \leq \theta \leq 2 \pi) .\) For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. $$ 2 \cos ^{2} \theta-\cos \theta=1 $$

The snowmobile sales \(S\) (in units) at a dealership are modeled by \(S=58.3+32.5 \cos \frac{\pi t}{6}\) where \(t\) is the time in months, with \(t=1\) corresponding to January \(1 .\) (a) Use a graphing utility to graph \(S\). (b) Will the sales exceed 75 units during any month? If so, during which month(s)?

find two values of \(\theta\) corresponding to each function. List the measure of \(\theta\) in radians \((0 \leq \theta \leq 2 \pi) .\) Do not use a calculator. $$ \text { (a) } \sec \theta=2 \quad \text { (b) } \sec \theta=-2 $$

sketch the graph of the function by hand. Use a graphing utility to verify your sketch. $$ y=\frac{3}{2} \cos \frac{2 x}{3} $$

solve the equation for \(\theta\) \((0 \leq \theta \leq 2 \pi) .\) For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. $$ \cos ^{2} \theta+\sin \theta=1 $$
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