Problem 64 - Calculus for Biology and Medicine

Question

Evaluate each integral. $$ \int \frac{1}{(x+1)\left(x^{2}+4\right)} d x $$

Step-by-Step Solution

Answer

Use partial fraction decomposition and integrate each term separately.

Step 1: Identify the Method

To solve this integral, we should use the technique of partial fraction decomposition. The given integral has a rational function where the denominator is a product of linear and quadratic factors, suitable for partial fraction decomposition.

Step 2: Set Up Partial Fraction Decomposition

Express the integrand \( \frac{1}{(x+1)(x^2+4)} \) as a sum of partial fractions. Let \[ \frac{1}{(x+1)(x^2+4)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 4} \] Multiply through by the denominator \((x+1)(x^2+4)\) to eliminate the fractions.

Step 3: Solve for Coefficients

Equate coefficients of the resulting equation:\[ 1 = A(x^2 + 4) + (Bx + C)(x+1) \]Expand and group like terms to solve for \(A\), \(B\), and \(C\).

Step 4: Solve for A

Substitute \(x = -1\) to find \(A\):\[ 1 = A((-1)^2 + 4) \Rightarrow A = \frac{1}{5} \]

Step 5: Substitute A to Solve for B and C

By substituting \(A = \frac{1}{5}\) into the expanded equation:\[ 1 = \frac{1}{5}(x^2 + 4) + (Bx + C)(x+1) \]Comparing coefficients for terms involving \(x^2\), \(x\), and constant terms allows us to solve for \(B\) and \(C\).

Step 6: Integrate Each Fraction

Decompose the original integral into simpler integrals:\[ \int \frac{1}{(x+1)(x^2+4)} \, dx = \int \frac{1}{5(x+1)} \, dx + \int \frac{Bx + C}{x^2 + 4} \, dx \]Integrate term by term.

Step 7: Integrate the Quotient of Constants

The integral \( \int \frac{1}{5(x+1)} \, dx \) becomes:\[ \frac{1}{5} \ln |x+1| + C_1 \]

Step 8: Integrate the Quadratic Expression

For the integral \( \int \frac{Bx + C}{x^2 + 4} \, dx \), separate into two parts and use substitution and arctangent identities if necessary:The integral will give terms of different forms based on \(B\) and \(C\).

Step 9: Write the Full Integrated Expression

Combine results from Steps 7 and 8 to write down the full expression for the indefinite integral, including constants of integration.

Key Concepts

Rational Function Integration

Integrating rational functions is an essential skill in calculus. These integrals consist of a polynomial divided by another polynomial. The key idea is to simplify these integrals through techniques like partial fraction decomposition.
This method is particularly useful for rational functions with complex denominators, combining linear or quadratic terms.

• First, check if the degree of the polynomial in the numerator is less than that of the denominator. If not, perform polynomial long division.
• Next, consider factorizing the denominator, if possible.
• Once simplified, utilize partial fraction decomposition where applicable, to breakdown more complicated rational functions into simpler terms for integration.

Remember, making complex rational functions more approachable simplifies the integration process.

Linear and Quadratic Factors

Partial fraction decomposition often involves breaking complicated fractions into a sum of simpler fractions. These simplifications help in straightforward integration.

• Linear factors: If the denominator contains factors like \(x + a\), use coefficients for simple fractions.
• Quadratic factors: When the denominator includes irreducible quadratic terms (like \(x^2 + b\)), decompose using linear numerators, such as \(Bx + C\).

Partial fractions are constructed with these coefficients to match the original function, simplifying the process to later solve for each coefficient individually.
Understanding and identifying these factors properly leads to a seamless integration.

Coefficients Solving

After setting up the partial fraction decomposition, it's time to solve for coefficients. This entails equating the original fraction to its decomposed fractions and solving algebraically.

• Clear the denominators by multiplying each term by the original denominator.
• Expand and equate coefficients on both sides, matching powers of \(x\).
• Solve the resulting system of equations, often by substituting convenient values of \(x\) to simplify.

Finding the right values for \(A\), \(B\), and \(C\) is critical, as these form the basis of the simpler integrals obtained from decomposition.