Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 899

\(\int(x-1) e^{-x} d x=\square+c\)

Short Answer

Expert verified
The integral of \((x-1)e^{-x} dx\) can be found using integration by parts, with \(u = x-1\) and \(dv = e^{-x} dx\). We then compute \(du = dx\) and \(v = -e^{-x}\) and apply the integration by parts formula, resulting in the final answer: \(-e^{-x}(x-1) - e^{-x} + c\), where \(c\) is the constant of integration.

Step by step solution

01

Understanding the Integration by Parts Formula

The integration by parts formula is given by \(\int u \ dv = u \cdot v - \int v \ du\), where \(u\) and \(dv\) are parts of the integrand that are selected in a way that makes the right-hand side simpler than the original integral. For the given problem, we need to carefully choose \(u\) and \(dv\) to simplify our integral.
02

Choose \(u\) and \(dv\) Appropriately

In this instance, let \(u = x - 1\) and \(dv = e^{-x} dx\). The reason behind this choice is that the derivative of \(x - 1\) will be simplified (i.e., the exponent of \(x\) will reduce), and the integral of \(e^{-x} dx\) is straightforward to find.
03

Compute \(du\) and \(v\)

After choosing \(u\) and \(dv\), find \(du\) (the derivative of \(u\)) and \(v\) (which is the integral of \(dv\)). This results in \(du = dx\) and \(v = -e^{-x}\).
04

Apply the Integration by Parts Formula

Substituting our values of \(u\), \(dv\), \(du\) and \(v\) into the integration by parts formula, we get \(\int (x - 1)e^{-x} dx = (x - 1) (-e^{-x}) - \int -e^{-x} dx\).
05

Simplify the Integral

We then simplify the expression to get: \(-e^{-x}(x - 1) + \int e^{-x} dx\).
06

Evaluate Remaining Integral

The integral \(\int e^{-x} dx\) is easy to compute; it is \(-e^{-x}\), because the derivative of \(-e^{-x}\) gives \(e^{-x}\).
07

Final Result

Substituting back in, we get: \(-e^{-x}(x - 1) - e^{-x} + c\), where \(c\) is the constant of integration. Thus, the integral of \((x - 1)e^{-x}\) is \(-e^{-x}(x - 1) - e^{-x} + c\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \(\int[\\{(-\sin x+\cos x) d x\\}\) \(/\left\\{(\sin x+\cos x) \sqrt{\left. \left.\left(\sin x \cos x+\sin ^{2} x \cos ^{2} x\right)\right\\}\right]}=\operatorname{cosec}^{-1}[f(x)]+c\right.\) then \(\mathrm{f}(\mathrm{x})=\) (a) \(\sin 2 \mathrm{x}+1\) (b) \(1-\sin 2 x\) (c) \(\sin 2 x-1\) (d) \(\cos 2 \mathrm{x}+1\)

If \(\int[(1+\cos 8 \mathrm{x}) /(\cot 2 \mathrm{x}-\tan 2 \mathrm{x})] \mathrm{dx}=\mathrm{Acos} 8 \mathrm{x}+\mathrm{C}\) then \(\mathrm{A}=\) (a) \(\overline{(1 / 16)}\) (b) \(-(1 / 8)\) (c) \(-(1 / 16)\) (d) \((1 / 8)\)

if \(\int\left[\left(4 \mathrm{e}^{\mathrm{x}}+6 \mathrm{e}^{-\mathrm{x}}\right) /\left(9 \mathrm{e}^{\mathrm{x}}-4 \mathrm{e}^{-\mathrm{x}}\right)\right] \mathrm{dx}=\mathrm{Ax}+\mathrm{B} \log \left(9 \mathrm{e}^{2 \mathrm{x}}-4\right)+\mathrm{c}\) then \(\mathrm{A}, \mathrm{B}=\) (a) \((3 / 2),[(-35) /(36)]\) (b) \(-(3 / 2),[(-35) /(36)]\) (c) \(-(3 / 2),(35 / 36)\) (d) \((3 / 2),(35 / 36)\)

\(\int e^{2 x}(1+\tan x)^{2} d x=\) \(+c\) (a) \(\tan \mathrm{e}^{\mathrm{x}}\) (b) \(\tan \mathrm{x} \mathrm{e}^{2 \mathrm{x}}\) (c) \(\tan (\mathrm{x} / 2) \mathrm{e}^{\mathrm{x}}\) (d) \(\tan (\mathrm{x} / 2) \mathrm{e}^{-\mathrm{x}}\)

\(\int[(\tan \mathrm{x}) /\\{\sqrt{(\cos \mathrm{x})\\}]}=\) \(+c\) (a) \([(+2) / \sqrt{(\cos x)}]\) (b) \(-[1 / \sqrt{(\cos x)]}\) (c) \([(-2) /\\{3 \sqrt{(\cos x})\\}]\) (d) \([(-3) /\\{2 \sqrt{(\cos x})\\}]\)
See all solutions

Recommended explanations on English Textbooks

Listening and Speaking

Read Explanation

Textual Analysis

Read Explanation

Pragmatics

Read Explanation

Argumentative Essay

Read Explanation

Global English

Read Explanation

Research and Composition

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.

Sign-up for free