Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Use partial fractions to find the indefinite integral. $$ \int \frac{4 x^{2}+2 x-1}{x^{3}+x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(4 \ln|x| - 2/x - 1/(2x^2) + C\).

Step 1: Decompose into partial fractions

To transform the fraction into the sum of simpler fractions, perform the following operations: \( \frac{4x^2 + 2x - 1}{x^3 + x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} \) Multiply both sides by \(x^3 + x^2\) to clear the fraction: \(4x^2 + 2x - 1 = Ax^2 + Ax + Bx + C\) Comparing coefficients from both sides gives three simultaneous equations: 1) \(A = 4\), 2) \(B + A = 2\), 3) \(C + B = -1\). Solving these equations gives \(A = 4\), \(B = -2\), \(C = 1\).

Step 2: Substitute and Integrate

Substitute \(A = 4\), \(B = -2\), \(C = 1\) into the partial fractions. We then get: \( \frac{4x^2 + 2x - 1}{x^3 + x^2} = 4/x - 2/x^2 + 1/x^3 \). Now integrating each term separately: \( \int{4/x\, dx} - \int{2/x^2\, dx} + \int{1/x^3\, dx} \). The first term gives \(4 \ln|x|\), the second gives \(2/x\) and the third gives \(-1/(2x^2)\).

Step 3: Consolidate the Result

Combine the results from the previous integrations for the final result: \(4 \ln|x| - 2/x - 1/(2x^2) + C\). This is the indefinite integral for the original function, where \(C\) is the constant of integration.