/*! 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 67 Use the reduction formulas in a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 67

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int x^{3} e^{2 x} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int x^{3} e^{2 x} d x$$ using reduction formulas. Answer: The integral is evaluated as $$x^3 e^{2x} (\frac{1}{8} - \frac{3}{8} x + \frac{3}{4} x^2 - \frac{3}{8} x^3) + C$$ where \(C\) is the constant of integration.

Step by step solution

01

Identify the parts for integration by parts

First, we need to identify the two parts for integration by parts. We will take the reduction formula found in the table of integrals which states that: $$\int x^n e^{ax} dx = \frac{1}{a^n} x^n e^{ax} - \frac{n}{a} \int x^{n-1} e^{ax} dx$$ In our case, we have \(n = 3\), \(a = 2\), and \(x^{3} e^{2x}\). You will see that we need to apply the formula until the exponent of x decreases.
02

Apply the reduction formula to \(n = 3\)

Applying the reduction formula to \(\int x^{3} e^{2x} dx\), we get: $$\int x^{3} e^{2x} dx = \frac{1}{8} x^3 e^{2x} - \frac{3}{2} \int x^{2} e^{2x} dx$$
03

Apply the reduction formula to \(n = 2\)

Now apply the formula to the new integral \(\int x^{2} e^{2x} dx\): $$\int x^{2} e^{2x} dx = \frac{1}{4} x^2 e^{2x} - \frac{2}{2} \int x^1 e^{2x} dx$$
04

Apply the reduction formula to \(n = 1\)

Lastly, apply the formula to the simplest integral \(\int x^1 e^{2x} dx\): $$\int x^1 e^{2x} dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} dx$$
05

Evaluate remaining integral

Now evaluate the remaining integral \(\int e^{2x} dx\): $$\int e^{2x} dx = \frac{1}{2} e^{2x} + C$$
06

Substitute steps 4, 3, and 2 to find the final solution

Replace the simplified integral in step 4: $$\int x^1 e^{2x} dx = \frac{1}{2} x e^{2x} - \frac{1}{2} (\frac{1}{2} e^{2x} + C)$$ Now substitute this into step 3: $$\int x^2 e^{2x} dx = \frac{1}{4} x^2 e^{2x} - (\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x}+ C_1)$$ Finally, substitute this into step 2: $$\int x^{3} e^{2 x} d x = \frac{1}{8} x^3 e^{2x} - \frac{3}{2} (\frac{1}{4} x^2 e^{2x} - (\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x}+ C_1)) + C_2$$
07

Simplify the final solution

Now, simplify the expression above to obtain the final solution: $$\int x^{3} e^{2 x} d x = \frac{1}{8} x^3 e^{2x} - \frac{3}{8} x^2 e^{2x} + \frac{3}{4} xe^{2x} - \frac{3}{8} e^{2x} + C = x^3 e^{2x} (\frac{1}{8} - \frac{3}{8} x + \frac{3}{4} x^2 - \frac{3}{8} x^3) + C$$

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

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int \frac{4 x^{6}}{x^{4}+3 x^{2}} d x\), the first step is to find the partial fraction decomposition of the integrand. b. The easiest way to evaluate \(\int \frac{6 x+1}{3 x^{2}+x} d x\) is with a partial fraction decomposition of the integrand. c. The rational function \(f(x)=\frac{1}{x^{2}-13 x+42}\) has an irreducible quadratic denominator. d. The rational function \(f(x)=\frac{1}{x^{2}-13 x+43}\) has an irreducible quadratic denominator.

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$

Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 2 and an illustration.

$$\text { Evaluate } \int \frac{d x}{1+\sin x+\cos x} \text { using the }$$ substitution \(x=2 \tan ^{-1} \theta .\) The identities \(\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\) are helpful.

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{\left(e^{x}+e^{-x}\right)^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

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.