Problem 28 $$ \text { In Problems } , \te... [FREE SOLUTION]

Chapter 7: Problem 28

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{2 x-1}{(x+4)(x+1)} d x $$

Short Answer

Expert verified

The integral evaluates to $ \ln \left( \frac{(x+4)^3}{x+1} \right) + C $.

Step by step solution

Decompose the Fraction

To evaluate the integral, we first need to decompose the fraction $ \frac{2x-1}{(x+4)(x+1)} $ into partial fractions. Assume $ \frac{2x-1}{(x+4)(x+1)} = \frac{A}{x+4} + \frac{B}{x+1} $.

Find Coefficients A and B

Multiply both sides by $(x+4)(x+1)$ to clear the denominators: \[ 2x - 1 = A(x+1) + B(x+4) \] Expand and combine like terms: \[ 2x - 1 = Ax + A + Bx + 4B \] Combine the $x$ terms: \[ 2x - 1 = (A+B)x + (A+4B) \] Set up equations by comparing coefficients: 1) $ A + B = 2 $ 2) $ A + 4B = -1 $

Solve for A and B

Solve the system of equations: From equation 1): $ A + B = 2 $ From equation 2): $ A + 4B = -1 $Subtract equation 1 from equation 2:\[ A + 4B - (A + B) = -1 - 2 \]\[ 3B = -3 \]\[ B = -1 \]Substitute $ B = -1 $ into equation 1:\[ A - 1 = 2 \]\[ A = 3 \]

Set Up the Integral with Partial Fractions

Substitute $ A $ and $ B $ back into the partial fractions: \[ \int \left( \frac{3}{x+4} - \frac{1}{x+1} \right) dx \] This allows us to integrate each term separately.

Integrate Each Term

Integrate each term separately: \[ \int \frac{3}{x+4} \, dx - \int \frac{1}{x+1} \, dx \] The integrals are: 1) $ \int \frac{3}{x+4} \, dx = 3 \ln |x+4| + C_1 $ 2) $ \int \frac{1}{x+1} \, dx = \ln |x+1| + C_2 $ Combine the results: \[ 3 \ln |x+4| - \ln |x+1| + C \] where $ C = C_1 - C_2 $.

Final Result

Combine the logarithms using properties of logarithms: \[ = \ln \left( \frac{(x+4)^3}{x+1} \right) + C \] Ensure the expression is simplified and includes the constant of integration.

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Key Concepts

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

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions. These simpler fractions can be integrated more easily.
In our problem, we started by taking the fraction $ \frac{2x-1}{(x+4)(x+1)} $ and expressing it as the sum of two simpler fractions: $ \frac{A}{x+4} + \frac{B}{x+1} $.

Step 1: Assume that the original fraction equals the sum of simpler fractions.
Step 2: Eliminate the denominators by multiplying both sides by $(x+4)(x+1)$.
Step 3: You get an equation in terms of $x$ with combined like terms.

By comparing coefficients, you set up equations to solve for constants $A$ and $B$. This process allows us to simplify and integrate later.

Definite and Indefinite Integrals

Once you have broken down a fraction via partial fraction decomposition, you can focus on integrating those fractions. Integrals can be classified into two types: definite and indefinite.
In this exercise, we focus on indefinite integrals. An indefinite integral represents a family of functions and includes a constant of integration $C$.

Indefinite Integrals: These do not have specified limits and represent antiderivatives.
Definite Integrals: These have limits and represent the area under a curve between two points.

For example, integrating $ \frac{3}{x+4} $ results in $ 3 \ln |x+4| + C_1 $, while integrating $ \frac{1}{x+1} $ results in $ \ln |x+1| + C_2 $. The constant $C$ is necessary as the antiderivative isn't a single function.

Logarithmic Integration

Logarithmic integration is used for integrals of the form $ \int \frac{1}{x+c} \, dx $, which result in a natural logarithm. This concept was applied after setting up the partial fractions.

The integration of $ \frac{3}{x+4} $ produces $ 3 \ln |x+4| $.
The integration of $ \frac{1}{x+1} $ gives $ \ln |x+1| $.

By the properties of logarithms, specifically $ \ln a - \ln b = \ln \left( \frac{a}{b} \right) $, we can combine the results:
The final answer becomes $ \ln \left( \frac{(x+4)^3}{x+1} \right) + C $. This step simplifies our expression by using logarithmic properties to combine like terms, maintaining simplicity and clarity.

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91影视

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{2 x-1}{(x+4)(x+1)} d x $$

Short Answer

Step by step solution

Decompose the Fraction

Find Coefficients A and B

Solve for A and B

Set Up the Integral with Partial Fractions

Integrate Each Term

Final Result

Key Concepts

Partial Fraction Decomposition

Definite and Indefinite Integrals

Logarithmic Integration

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