6. Given that \( \int_{1}^{5} f(x) d x=8, \int_{4}^{5} f(x) d x=5 \) and \( \int_{1}^{8} f(x) d x=15 \). Show that \( \int_{4}^{8} f(x) d x=12 \).

To show that \( \int_{4}^{8} f(x) \, dx = 12 \), we can use the **additivity property of definite integrals**, which states that if you have adjacent intervals, the integral over the combined interval is the sum of the integrals over each part. Given: 1. \( \int_{1}^{5} f(x) \, dx = 8 \) 2. \( \int_{4}^{5} f(x) \, dx = 5 \) 3. \( \int_{1}^{8} f(x) \, dx = 15 \) **Step 1:** Express \( \int_{1}^{8} f(x) \, dx \) as the sum of two integrals: \[ \int_{1}^{8} f(x) \, dx = \int_{1}^{5} f(x) \, dx + \int_{5}^{8} f(x) \, dx \] **Step 2:** Substitute the known values: \[ 15 = 8 + \int_{5}^{8} f(x) \, dx \] \[ \int_{5}^{8} f(x) \, dx = 15 - 8 = 7 \] **Step 3:** Now, express \( \int_{4}^{8} f(x) \, dx \) as the sum of two integrals: \[ \int_{4}^{8} f(x) \, dx = \int_{4}^{5} f(x) \, dx + \int_{5}^{8} f(x) \, dx \] **Step 4:** Substitute the known values: \[ \int_{4}^{8} f(x) \, dx = 5 + 7 = 12 \] Therefore, \[ \int_{4}^{8} f(x) \, dx = 12 \] **Answer:** \(\int_{4}^{8} f(x)\,dx = \int_{4}^{5}f(x)\,dx + \int_{5}^{8}f(x)\,dx = 5 + 7 = 12\)