/*! 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 58 Complete the square in the denom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 58

Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+2 x+5} d x $$

Short Answer

Expert verified
The integral evaluates to \(\frac{1}{2} \tan^{-1}\left(\frac{x+1}{2}\right) + C\).

Step by step solution

01

Identify the Quadratic Expression

The expression in the denominator is a quadratic: \(x^2 + 2x + 5\). Our goal is to rewrite it by completing the square.
02

Complete the Square in the Quadratic

To complete the square, start with the expression \(x^2 + 2x\). Add and subtract the square of half the coefficient of \(x\), which is \(1\): \(x^2 + 2x + 1 - 1 + 5\). This simplifies to \((x+1)^2 + 4\).
03

Rewrite the Integral

Substitute the completed square form back into the integral: \(\int \frac{1}{(x+1)^2 + 4} \, dx\).
04

Substitute for Simplicity

Let \(u = x+1\), so \(x = u-1\) and \(du = dx\). The integral becomes \(\int \frac{1}{u^2 + 4} \, du\).
05

Recognize the Standard Form

The integral is of the form \(\int \frac{1}{u^2 + a^2} \, du\), with \(a^2 = 4\) or \(a = 2\). The antiderivative is \(\frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C\).
06

Apply the Antiderivative Formula

Using the formula, the integral is \(\frac{1}{2} \tan^{-1}\left(\frac{u}{2}\right) + C\).
07

Substitute Back to Original Variable

Replace \(u\) with \(x+1\) to get the final solution: \(\frac{1}{2} \tan^{-1}\left(\frac{x+1}{2}\right) + C\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a powerful algebraic technique mainly used in integration to simplify complex expressions. When you encounter a quadratic expression, it can be rewritten in a form that is squareroot friendly.
For the quadratic expression in the exercise, \(x^2 + 2x + 5\), you aim to express it as a perfect square plus a constant.

Here鈥檚 a simple guide to practice:
  • Isolate the quadratic and linear terms: Take \(x^2 + 2x\) separately.
  • Add and subtract \((\frac{b}{2})^2\): Where \(b\) is the coefficient of \(x\). Here, \(2\), so \((\frac{2}{2})^2 = 1\). You add and also subtract this value to keep the expression unchanged: \(x^2 + 2x + 1 - 1 + 5\).
  • Rewrite: Now, it is easier to see \((x+1)^2 + 4\).
This makes using other integration techniques more straightforward, such as trigonometric substitution.
Trigonometric Substitution
Trigonometric substitution is a technique used to evaluate integrals that involve square roots or quadratic expressions. After completing the square, it's common to implement this method to transform the integral
into something that matches a trigonometric identity, simplifying the integration process.

Here鈥檚 how you utilize trigonometric substitution based on the completed square:
  • With the expression \((x+1)^2 + 4\), or written as \(u^2 + 4\) after substitution, recognize it is in the form \(u^2 + a^2\).
  • Substitute: Use \(u = x+1\) and choose \(u = a \tan(\theta)\) where \(a = 2\) to fit the identity \(1 + \tan^2(\theta) = \sec^2(\theta)\).
  • Simplify: The differential \(du = a \sec^2(\theta) d\theta\) can replace \(du\), facilitating the integration of trigonometric functions.
Using trigonometric identities like these simplifies the antiderivative finding process, making it possible to arrive at the exact result quickly.
Indefinite Integral
An indefinite integral, sometimes referred to as an antiderivative, represents a fundamental concept in calculus. Unlike a definite integral, which calculates a precise area under the curve, an indefinite integral offers a family of functions representing all possible antiderivatives of the original function.

To unpack the process with the example
  • After completing the square and substituting, we encounter the standard formula: \( \int \frac{1}{u^2 + a^2} \, du\).
  • This aligns with the recognizable antiderivative formula: \( \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C\), where \(C\) indicates a constant of integration.
  • The substitution shifts back after integration to keep expressions in terms of the original variable \(x\).
Mastering indefinite integrals involves recognizing form patterns you translate into known antiderivative formulas. This is essential for simplifying expressions and finding solutions to calculus problems efficiently.

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

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\cos x, a=\frac{\pi}{2}, n=3 ; x=\frac{\pi}{3} $$

Use the Table of Integrals to compute each integral. $$ \int \sqrt{x^{2}+16} d x $$

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=1 /(1+x), x=0.2, \text { error }<10^{-2} $$

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=e^{x}, x=2, \text { error }<10^{-3} $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{-1}^{1} \sqrt{x+1} d x\) (a) \(n=10\) (b) \(n=30\).
See all solutions

Recommended explanations on Biology Textbooks

Ecosystems

Read Explanation

Biological Organisms

Read Explanation

Communicable Diseases

Read Explanation

Organ Systems

Read Explanation

Cell Communication

Read Explanation

Plant Biology

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.