Problem 28 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( \frac{2x-1}{(x+4)(x+1)} = \frac{A}{x+4} + \frac{B}{x+1} \).

Step 2: 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) \( A + 4B = -1 \)

Step 3: 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:\[ A - 1 = 2 \]\[ A = 3 \]

Step 4: 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.

Step 5: 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 \( C = C_1 - C_2 \).

Step 6: 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.

Key Concepts

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.