Let's solve each definite integral step by step. --- ### **a.** \(\int_{3}^{4} (3e^{2x} - 7x^4 - 15x + 17) \, dx\) #### **Step 1: Integrate each term separately** 1. \(\int 3e^{2x} \, dx = 3 \int e^{2x} dx = 3 \cdot \frac{1}{2} e^{2x} = \frac{3}{2} e^{2x}\) 2. \(\int -7x^4 \, dx = -7 \int x^4 dx = -7 \cdot \frac{1}{5} x^5 = -\frac{7}{5} x^5\) 3. \(\int -15x \, dx = -15 \cdot \frac{1}{2} x^2 = -\frac{15}{2} x^2\) 4. \(\int 17 \, dx = 17x\) So, the indefinite integral is: \[ F(x) = \frac{3}{2}e^{2x} - \frac{7}{5}x^5 - \frac{15}{2}x^2 + 17x \] #### **Step 2: Evaluate from 3 to 4** \[ F(4) = \frac{3}{2}e^{8} - \frac{7}{5}(4^5) - \frac{15}{2}(4^2) + 17 \cdot 4 \] \[ F(3) = \frac{3}{2}e^{6} - \frac{7}{5}(3^5) - \frac{15}{2}(3^2) + 17 \cdot 3 \] Calculate values: - \(4^5 = 1024\) - \(3^5 = 243\) - \(4^2 = 16\) - \(3^2 = 9\) \[ F(4) = \frac{3}{2}e^{8} - \frac{7}{5}(1024) - \frac{15}{2}(16) + 68 \] \[ = \frac{3}{2}e^{8} - 1433.6 - 120 + 68 \] \[ = \frac{3}{2}e^{8} - 1485.6 \] \[ F(3) = \frac{3}{2}e^{6} - \frac{7}{5}(243) - \frac{15}{2}(9) + 51 \] \[ = \frac{3}{2}e^{6} - 340.2 - 67.5 + 51 \] \[ = \frac{3}{2}e^{6} - 356.7 \] So, \[ \int_{3}^{4} (3e^{2x} - 7x^4 - 15x + 17) dx = F(4) - F(3) \] \[ = \left( \frac{3}{2}e^{8} - 1485.6 \right) - \left( \frac{3}{2}e^{6} - 356.7 \right) \] \[ = \frac{3}{2}(e^{8} - e^{6}) + ( -1485.6 + 356.7 ) \] \[ = \frac{3}{2}(e^{8} - e^{6}) - 1128.9 \] **Final Answer for (a):** \[ \boxed{ \frac{3}{2}(e^{8} - e^{6}) - 1128.9 } \] --- ### **b.** \(\int_{1}^{5} \left( \sqrt{x - \frac{1}{x}} + 1 \right) dx\) #### **Step 1: Split the integral** \[ \int_{1}^{5} \left( \sqrt{x - \frac{1}{x}} + 1 \right) dx = \int_{1}^{5} \sqrt{x - \frac{1}{x}} dx + \int_{1}^{5} 1 dx \] The second term is easy: \[ \int_{1}^{5} 1 dx = 5 - 1 = 4 \] #### **Step 2: Deal with the first term** Let’s let \( u = x - \frac{1}{x} \implies du = 1 + \frac{1}{x^2} dx \). But this substitution does **not** simplify the integral easily. This integral does not have an elementary closed form, so we can only express the answer as: \[ \int_{1}^{5} \sqrt{x - \frac{1}{x}} dx + 4 \] Or, approximate numerically: Let’s try values at midpoint, or use numerical integration (Simpson's Rule, Trapezoidal, etc). Alternatively, **final answer in exact form:** \[ \boxed{ \int_{1}^{5} \sqrt{x - \frac{1}{x}} \, dx + 4 } \] If you want an approximate value, we can estimate with a calculator or numerical integration: --- #### **Numerical Approximation (using calculator)** Let’s estimate: - At \(x=1\), \(\sqrt{1-1}=0\) - At \(x=5\), \(\sqrt{5-1/5} = \sqrt{4.8} \approx 2.1909\) Let's estimate the area under the curve with the average value: \[ \text{Average value} \approx \frac{0 + 2.19}{2} = 1.095 \] \[ \text{Multiply by interval length: } (5-1) \cdot 1.095 = 4.38 \] Add to 4: \[ 4.38 + 4 \approx 8.38 \] **So, the approximate value is:** \[ \boxed{8.4} \] But the **exact answer** is: \[ \boxed{ \int_{1}^{5} \sqrt{x - \frac{1}{x}} dx + 4 } \] --- ## **Summary of Answers:** ### a. \[ \boxed{ \frac{3}{2}(e^{8} - e^{6}) - 1128.9 } \] ### b. \[ \boxed{ \int_{1}^{5} \sqrt{x - \frac{1}{x}} dx + 4 } \] or approximately \[ \boxed{ 8.4 } \]