/*! 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 37 Evaluate the integral. \( \disp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 37

Evaluate the integral. \( \displaystyle \int \frac{x^2 - 3x + 7}{(x^2 - 4x + 6)^2}\ dx \)

Short Answer

Expert verified
Use substitution with \( u = x^2 - 4x + 6 \) and integrate each part separately.

Step by step solution

01

Recognize the Type of Integral

The given integral is of the form \( \int \frac{P(x)}{Q(x)^n} \ dx \), with \( P(x) = x^2 - 3x + 7 \) and \( Q(x) = x^2 - 4x + 6 \), which suggests using integration by parts or simplification by substitution.
02

Simplify the Denominator

The denominator \( x^2 - 4x + 6 \) can potentially be simplified by completing the square: \( x^2 - 4x + 6 = (x - 2)^2 + 2 \). This helps to identify substitutions that can simplify the integration.
03

Perform Substitution

Let \( u = x^2 - 4x + 6 \), then \( \frac{du}{dx} = 2x - 4 \), or \( du = (2x - 4)dx \). Rearrange to solve for dx in terms of du: \( dx = \frac{du}{2x - 4} \).
04

Express the Integral in Terms of u

Using the substitutions from the previous steps, rewrite the integral as \( \int \frac{x^2 - 3x + 7}{u^2} \cdot \frac{du}{2x - 4} \). Now, express the numerator \( x^2 - 3x + 7 \) in terms of u and x.
05

Simplifying Further in Terms of x

Notice that \( x^2 = u + 4x - 6 \), and simplify the numerator in terms of x. Substitute into the integral: \( x^2 - 3x + 7 = (u + 4x - 6) - 3x + 7 \) becomes \( u + x + 1 \).
06

Separate the Integral

The integral now separates into simpler parts: \( \int \frac{u + x + 1}{u^2} \cdot \frac{du}{2x - 4} \). Divide each term and express the integral as several simpler integrals if possible.
07

Simplify and Integrate Each Term Separately

Evaluate each term separately, focusing primarily on simplifying terms involving x by using \( x = 2 \pm \sqrt{u - 2} \) based on the substitution made. Note that some terms will integrate to standard forms such as \( \int u^{-2} du \).
08

Combine the Results and Back-Substitute

Combine the results of each term's integration and substitute back using the original terms from the substitution \( u = x^2 - 4x + 6 \) to express the answer in terms of x.

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.

Integration by Parts
Integration by parts is a powerful technique for tackling integrals involving products of functions. It stems from the product rule of differentiation.
It's particularly useful when dealing with integrals of the form \( \int u \, dv \).
To apply integration by parts:
  • Select parts from the integrand: choose \( u \) and \( dv \) such that \( du \) and \( v \) are easy to find.
  • Use the formula \( \int u \, dv = uv - \int v \, du \).
  • Integrate the new integral \( \int v \, du \) that results from the transformation.
This technique is instrumental in tackling complex integrations, allowing simplification by rearranging the terms. For polynomial integrals where direct approaches fail, integration by parts can aid by breaking down expressions and reducing the order of powers incrementally.
Substitution Method
The substitution method, often called \( u \)-substitution, is a strategy for simplifying integrals by changing variables.
We usually use this technique when recognizing a certain pattern within the integral that resembles a derivative within another function.
Here's how to utilize substitution:
  • Identify a function \( u \) inside the integral whose derivative is also present or can be rearranged to fit the integral.
  • Set \( u = f(x) \), and calculate \( du = f'(x)dx \).
  • Replace all instances of \( f(x) \) and \( dx \) in the original integral with \( u \) and \( du \).
  • Integrate using \( u \), then convert back to the original variable by substituting \( x \) back in.
This technique simplifies the integrand into a more manageable form, making it easier to integrate. In our exercise, setting \( u = x^2 - 4x + 6 \) was a crucial step that transformed the integrand into a simpler structure.
Completing the Square
Completing the square is a handy method when dealing with quadratic expressions in integrals.
It transforms a quadratic polynomial into a perfect square plus or minus a constant, which can simplify substitution.
To complete the square for a quadratic \( ax^2 + bx + c \):
  • Rearrange terms and factor out any leading coefficients of \( x^2 \).
  • Calculate \((b/2)^2\) and add and subtract it within the expression.
  • Rewrite the quadratic as \((x - h)^2 + k \), where \( h \) and \( k \) are constants determined from the previous step.
In integrals, completing the square can reveal clearer substitution paths or lead to recognizably simpler integrands, such as trigonometric forms. In our problem, rewriting \( x^2 - 4x + 6 \) as \((x-2)^2 + 2\) made u\-substitution more straightforward.
Polynomial Integrals
Polynomial integrals involve the integration of algebraic expressions made up of sums of powers of \( x \).
These are some of the most straightforward integrals, using basic rules:
  • The power rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), applicable when \( n eq -1 \).
  • Each term in the polynomial is integrated separately and added together.
In our exercise, simplifying the expression \( x^2 - 3x + 7 \) into manageable components via substitutions, revealed underlying polynomial structures which could be integrated using standard rules. The key is recognizing when polynomials can be decomposed or used directly in standard integral forms, enabling a straightforward path to the solution.

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

(a) Show that \( \displaystyle \int_{-\infty}^\infty x\ dx \) is divergent. (b) Show that $$ \lim_{t\to\infty} \int_{-t}^t x\ dx = 0 $$ This shows that we can't define $$ \int_{-\infty}^\infty f(x)\ dx = \lim_{t\to\infty} \int_{-t}^t f(x)\ dx $$

Find the value of the constant \( C \) for which the integral $$ \displaystyle \int_0^\infty \left (\frac{1}{\sqrt{x^2 + 4}} - \frac{C}{x + 2} \right)\ dx $$ converges. Evaluate the integral for this value of \( C \).

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \( \displaystyle \int_{-\infty}^0 ze^{2z}\ dz \)

Show that if \( 0 \le f(t) \le Me^{at} \) for \( t \ge 0 \), where \( M \) and \( a \) are constants, then the Laplace transform \( F(s) \) exists for \( s > a \).

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \( \displaystyle \int_1^\infty \frac{1 + \sin^2 x}{\sqrt{x}}\ dx \)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete 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.