7:09 PM Sun 16 Mar - 35\% mylab.pearson.com Kurulus Osm... Can't Open P... CUNY Login The Evolutio... Do Homewor... Discord MAT 220-S25 Rasheed Larry 03/16/25 7:09 PM \( \equiv \) Homework: Partial Fractions, part 2 Question 3, 8.5.25 HW Score: 80\%, 4 of 5 points ® Points: 0 of 1 Save Question list Express the integrand as a sum of partial fractions and evaluate the integral. \[ \int \frac{24 s+24}{\left(s^{2}+1\right)(s-1)^{3}} d s \] Question 1 Question 2 \[ \int \frac{24 s+24}{\left(s^{2}+1\right)(s-1)^{3}} d s=\square \] Question 3 Question 4 Question 5 Help me solve this View an example Get more help Clear all Check answer

To express the integrand \(\frac{24s+24}{(s^2+1)(s-1)^3}\) as a sum of partial fractions, we start by setting up the equation: \[ \frac{24s+24}{(s^2+1)(s-1)^3} = \frac{As + B}{s^2 + 1} + \frac{C}{s - 1} + \frac{D}{(s - 1)^2} + \frac{E}{(s - 1)^3} \] Next, find coefficients \(A\), \(B\), \(C\), \(D\), and \(E\) by multiplying both sides by the denominator \((s^2 + 1)(s - 1)^3\) and simplifying. Once you have the right values, break down the integral, solve each term, and you'll have your answer! For a real-world application, integration using partial fractions often comes into play in electrical engineering, particularly in circuit design. When analyzing complex systems, engineers use integrals to determine the system's response to various inputs. Understanding how to break down functions into manageable pieces allows for easier calculations of voltage, current, or energy over time, ensuring that the systems designed are both efficient and effective!