Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 77
Evaluate the integrals in Exercises \(67-80\) $$ \int \frac{x^{2}+2 x-1}{x^{2}+9} d x $$
Short Answer
Expert verified
The integral evaluates to \( x + \ln |x^2 + 9| - \frac{10}{3} \arctan\left(\frac{x}{3}\right) + C \).
Step by step solution
01
Understand the Problem
We are given the integral \( \int \frac{x^{2}+2x-1}{x^{2}+9} \, dx \). Our goal is to evaluate this definite integral.
02
Simplify the Integrand
We can rewrite the numerator \(x^2 + 2x - 1\) to match the denominator \(x^2 + 9\):\(x^2 + 2x - 1 = (x^2 + 9) - 9 + 2x - 1 = (x^2 + 9) + (2x - 10)\).
03
Separate the Integral
Separate the integral into two parts:\[\int \frac{(x^2 + 9) + (2x - 10)}{x^2 + 9} \, dx = \int \frac{x^2 + 9}{x^2 + 9} \, dx + \int \frac{2x - 10}{x^2 + 9} \, dx \] This simplifies to:\[\int 1 \, dx + \int \frac{2x - 10}{x^2 + 9} \, dx \]
04
Integrate the Constant Term
The integral \(\int 1 \, dx \) simply results in \(x + C_1\), where \(C_1\) is a constant.
05
Solve the Remaining Integral
Let鈥檚 focus on \(\int \frac{2x - 10}{x^2 + 9} \, dx \). We can split this into two integrals:\[\int \frac{2x}{x^2 + 9} \, dx - \int \frac{10}{x^2 + 9} \, dx \].
06
Use Substitution for the First Part
Use the substitution \(u = x^2 + 9\), so \(du = 2x \, dx\). The integral becomes:\[ \int \frac{du}{u} = \ln |u| + C_2 = \ln |x^2 + 9| + C_2 \].
07
Recognize the Arctangent Formula for the Second Part
The integral \( \int \frac{10}{x^2 + 9} \, dx \) matches the form \( \int \frac{a}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C_3 \) with \(a = 3\). Hence:\[ \int \frac{10}{x^2 + 9} \, dx = \frac{10}{3} \arctan\left(\frac{x}{3}\right) + C_3 \].
08
Combine All Results
The integral is the sum of all evaluated parts:\[ x + \ln |x^2 + 9| - \frac{10}{3} \arctan\left(\frac{x}{3}\right) + C \]where \(C = C_1 + C_2 + C_3\).
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.
Definite Integrals
A definite integral evaluates the area under the curve of a given function between two points. Unlike indefinite integrals, which result in a general function plus a constant, definite integrals yield a numeric value, representing an area. This is calculated using limits. So, when you compute a definite integral, the boundaries of the interval ensure there's no need for an additional constant at the end.
- The notation for a definite integral is given by: \( \int_{a}^{b} f(x) \, dx \).
- Remember, the function \( f(x) \) represents the curve, and \( a \) and \( b \) are the lower and upper limits, respectively.
Substitution Method
The substitution method simplifies the process of integrating complex functions by transforming them into simpler forms.
- This approach involves substituting part of the integral with a new variable, making it easier to work with.
- For instance, if you have \( \int f(g(x))g'(x) \, dx \), you let \( u = g(x) \). As a result, \( du = g'(x) \, dx \). The integral then becomes \( \int f(u) \, du \).
Arctangent Formula
The arctangent formula is a specific integration technique dealing with integrands having the form \( \int \frac{a}{x^2 + a^2} \, dx \).
- The result of such an integral is given by \( \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C \), where \( C \) is the constant of integration.
- This formula is derived from recognizing the derivative of \( \arctan(x) \), which is \( \frac{1}{1+x^2} \), and adjusting it for the constant factor \( a \).
