**Problem 4: Integral Calculus**Evaluate the following integral:\[ \int \frac{-5x^2 - 16x + 28}{x^3 - 4x} \, dx \]This problem requires the evaluation of the integral of a rational function. The integrand is:\[ \frac{-5x^2 - 16x + 28}{x^3 - 4x} \]with respect to \(x\), denoted by \(dx\). The denominator can be factored into \(x(x^2 - 4)\), which further factors into \(x(x - 2)(x + 2)\).To evaluate this integral, one common approach is partial fraction decomposition. This technique involves expressing the integrand as a sum of simpler fractions whose denominators are the factors of \(x^3 - 4x\).Here’s a step-by-step outline for solving it:1. **Factor the Denominator:** \[ x^3 - 4x = x(x^2 - 4) = x(x - 2)(x + 2) \]2. **Set Up Partial Fractions:** Express the integrand as a sum of partial fractions: \[ \frac{-5x^2 - 16x + 28}{x(x - 2)(x + 2)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} \]3. **Solve for Coefficients (A, B, and C):** Multiply both sides by the common denominator \(x(x - 2)(x + 2)\) to find the values of A, B, and C.4. **Integrate Each Term Separately:** Integrate the simpler fractions.The final result will be the sum of these integrals, and it will include a constant of integration, \(C\).For detailed step-by-step integration procedures, factoring explanations, and partial fraction decomposition techniques, refer to the calculus section on rational functions and integrals.

Name: **Problem 4: Integral Calculus**Evaluate the following integral:\[ \i
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: **Problem 4: Integral Calculus**Evaluate the following integral:\[ \int \frac{-5x^2 - 16x + 28}{x^3 - 4x} \, dx \]This problem requires the evaluation of the integral of a rational function. The integrand is:\[ \frac{-5x^2 - 16x + 28}{x^3 - 4x} \]wi