/*! 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 44 Use integration by parts to deri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 44

Use integration by parts to derive the following reduction formulas. $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad \text { for } a \neq 0$$

Short Answer

Expert verified
Question: Find the reduction formula for the following integral using integration by parts: $$\int x^{n} e^{a x} d x$$ Answer: The reduction formula for the given integral is: $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad \text { for } a \neq 0$$

Step by step solution

01

Choose u and dv

Let's start by choosing our functions \(u\) and \(dv\). We'll choose: $$u = x^n$$ and $$dv = e^{ax} dx$$
02

Compute du and v

Now, we need to compute \(du\) and \(v\): $$du = \frac{d}{dx}(x^n)dx = n x^{n-1} dx$$ $$v = \int e^{ax} dx = \frac{1}{a} e^{ax}$$
03

Apply integration by parts formula

Insert \(u\), \(dv\), \(du\), and \(v\) into the integration by parts formula: $$\int u dv = uv - \int v du$$ $$\int x^n e^{ax} dx = x^n \cdot \frac{1}{a} e^{ax} - \int \left(\frac{1}{a} e^{ax}\right) {\cdot nx^{n-1}dx}$$
04

Simplify the expression

Finally, simplify the expression: $$\int x^n e^{ax} dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax} dx$$ So, the reduction formula for the given integral is: $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad \text { for } a \neq 0$$

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

Exact Simpson's Rule Prove that Simpson's Rule is exact (no error) when approximating the definite integral of a linear function and a quadratic function.

Let \(I_{n}=\int x^{n} e^{-x^{2}} d x,\) where \(n\) is a nonnegative integer. a. \(I_{0}=\int e^{-x^{2}} d x\) cannot be expressed in terms of elementary functions. Evaluate \(I_{1}\). b. Use integration by parts to evaluate \(I_{3}\). c. Use integration by parts and the result of part (b) to evaluate \(I_{5}\). d. Show that, in general, if \(n\) is odd, then \(I_{n}=-\frac{1}{2} e^{-x^{2}} p_{n-1}(x)\) where \(p_{n-1}\) is a polynomial of degree \(n-1\). e. Argue that if \(n\) is even, then \(I_{n}\) cannot be expressed in terms of elementary functions.

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{e^{x}}{\left(e^{x}-1\right)\left(e^{x}+2\right)} d x$$

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

A long, straight wire of length \(2 L\) on the \(y\) -axis carries a current \(I\). According to the Biot-Savart Law, the magnitude of the magnetic field due to the current at a point \((a, 0)\) is given by $$B(a)=\frac{\mu_{0} I}{4 \pi} \int_{-L}^{L} \frac{\sin \theta}{r^{2}} d y$$ where \(\mu_{0}\) is a physical constant, \(a>0,\) and \(\theta, r,\) and \(y\) are related as shown in the figure. a. Show that the magnitude of the magnetic field at \((a, 0)\) is $$B(a)=\frac{\mu_{0} I L}{2 \pi a \sqrt{a^{2}+L^{2}}}$$ b. What is the magnitude of the magnetic field at \((a, 0)\) due to an infinitely long wire \((L \rightarrow \infty) ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.