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To solve the integral \(\int_1^2 3x^5 \, dx\), follow these steps: 1. **Find the antiderivative:** The antiderivative of \(3x^5\) is \(\frac{3}{6}x^6 = \frac{1}{2}x^6\). 2. **Evaluate the definite integral:** \[ \int_1^2 3x^5 \, dx = \left[\frac{1}{2}x^6\right]_1^2 \] 3. **Calculate:** \[ = \frac{1}{2}(2^6) - \frac{1}{2}(1^6) \] \[ = \frac{1}{2}(64) - \frac{1}{2}(1) \] \[ = 32 - \frac{1}{2} \] \[ = 31.5 \] The answer is 31.5.