/*! 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 1 Derive a reduction formula and c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 1

Derive a reduction formula and compute the integral \(\int_{-1}^{0} x^{n} e^{x} d x,(n\) is a positive integer \()\).

Short Answer

Expert verified
Question: Determine the definite integral \(\int_{-1}^{0} x^{n} e^{x} dx\) for positive integer values of \(n\). Answer: \(\int_{-1}^{0} x^{n} e^{x} dx = 1 - \sum_{k=0}^{n}(-1)^{k} \binom{n}{k} e^{-1}\)

Step by step solution

01

Perform integration by parts

Let's choose \(u=x^n\) and \(dv=e^x dx\). Thus, \(du=nx^{n-1}dx\) and \(v=e^x\). Then the integration by parts formula, \(\int u dv = u v - \int v du\), gives us: \(\int x^{n} e^{x} dx = x^n e^x - n\int x^{n-1} e^x dx\).
02

Look for a pattern for the reduction formula

Observe that: \(I_n = \int x^{n} e^{x} dx = x^n e^x - nI_{n-1}\), where \(I_n = \int x^{n} e^{x} dx\). The pattern for the reduction formula can be expressed as: \(I_n = x^n e^x - nx^{n-1} e^x + n(n-1) I_{n-2}\). By continuing this process, we can see a pattern that leads to the reduction formula: \(I_n = e^x\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}x^{n-k}\).
03

Apply the reduction formula to the given definite integral

Now we apply this reduction formula to compute the desired integral: \(\int_{-1}^{0} x^{n} e^{x} dx = \left[e^x\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}x^{n-k}\right]_{-1}^{0}\) \(= \left[e^0\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}0^{n-k} - e^{-1}\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}(-1)^{n-k}\right]\) \(= 1 - \sum_{k=0}^{n}(-1)^{k} \binom{n}{k} e^{-1}\). This is the final expression for the integral \(\int_{-1}^{0} x^{n} e^{x} dx\).

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

Evaluate the following integrals: (i) \(\int_{-\infty}^{\infty} \frac{x d x}{x^{4}+1}\) (ii) \(\int_{0}^{1} \frac{\ln (1-x)}{x} \mathrm{dx}\) (iii) \(\int_{0}^{\infty} \frac{\mathrm{dx}}{(\mathrm{x}+1)(\mathrm{x}+2)}\) (iv) \(\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}, a, b>0 .\)

A function \(\mathrm{f}\) is defined for all real \(\mathrm{x}\) by the formula \(\mathrm{f}(\mathrm{x})=3+\int_{0}^{\mathrm{x}} \frac{1+\sin \mathrm{t}}{2+\mathrm{t}^{2}} \mathrm{dt}\). Without attempting to evaluate this integral, find a quadratic polynomial \(\mathrm{p}(\mathrm{x})=\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{2}\) such that \(\mathrm{p}(0)=\mathrm{f}(0), \mathrm{p}^{\prime}(0)=\mathrm{f}^{\prime}(0)\), and \(\mathrm{p}^{\prime \prime}(0)=\) f' \((0)\).

Prove the inequalities: (i) \(\int_{1}^{3} \sqrt{x^{4}+1} d x \geq \frac{26}{3}\)(iii) \(\frac{1}{17} \leq \int_{1}^{2} \frac{1}{1+x^{4}} \mathrm{dx} \leq \frac{7}{24}\).

A periodic function with period 1 is integrable over any finite interval. Also for two real numbers \(\mathrm{a}, \mathrm{b}\) and for two unequal non-zero postive integers \(\mathrm{m}\) and \(\mathrm{n}, \int_{a}^{a+1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{\mathrm{b}}^{b+\mathrm{m}} \mathrm{f}(\mathrm{x}) \mathrm{dx} .\) Calculate the value of \(\int_{\mathrm{m}}^{\mathrm{n}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\)

Let p be a polynomial of degree atmost 4 such that \(\mathrm{p}(-1)=\mathrm{p}(1)=0\) and \(\mathrm{p}(0)=1\). If \(\mathrm{p}(\mathrm{x}) \leq 1\) for \(x \in[-1,1]\), find the largest value of \(\int^{1} p(x) d x\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.