## Step 1: Express the Definite Integral as a Limit of Riemann Sums Given: \[ \int_{4}^{8} (x - 4\ln x)dx \] Divide \([4, 8]\) into \(n\) subintervals, each of width \(\Delta x = \frac{8-4}{n} = \frac{4}{n}\). The right endpoint for the \(i\)th subinterval: \(x_i = 4 + i\Delta x = 4 + \frac{4i}{n}\) So, the Riemann sum is: \[ \sum_{i=1}^{n} \left[ x_i - 4\ln(x_i) \right] \Delta x \] ### Explanation: - We split the interval into \(n\) equal parts. - The right endpoint formula is used because the question specifies right endpoints. --- ## Step 2: Write the Limit Representation So, the limit is: \[ \lim_{n \to \infty} \sum_{i=1}^{n} \left[ \left(4 + \frac{4i}{n}\right) - 4\ln\left(4 + \frac{4i}{n}\right)\right] \cdot \frac{4}{n} \] ### Explanation: - Substitute the expressions for \(x_i\) and \(\Delta x\) into the Riemann sum. - Take the limit as \(n \to \infty\) for the exact value. --- ## Final Answer (Riemann Sum Representation): \[ \boxed{ \lim_{n \to \infty} \sum_{i=1}^{n} \left[ \left(4 + \frac{4i}{n}\right) - 4\ln\left(4 + \frac{4i}{n}\right)\right] \cdot \frac{4}{n} } \] --- ## Step 1 (Indefinite Integral): Integrate Each Term Given: \[ \int \left( \cos x + \frac{1}{5}x \right) dx \] - \(\int \cos x \, dx = \sin x\) - \(\int \frac{1}{5}x \, dx = \frac{1}{5} \cdot \frac{x^2}{2} = \frac{1}{10}x^2\) ### Explanation: - Integrate term by term using basic integration rules. --- ## Step 2 (Include the Constant): Combine the results and add the constant \(C\): \[ \sin x + \frac{1}{10}x^2 + C \] ### Explanation: - All indefinite integrals should include the arbitrary constant \(C\). --- ## Final Answer (Indefinite Integral): \[ \boxed{\sin x + \frac{1}{10}x^2 + C} \]