Let \( \vec{r}(t)=\left\langle\sqrt[3]{t-4}, 4 e^{3 t}, \frac{2}{t-11}\right\rangle \). Evaluate the definite integral. \( \int_{12}^{31} \vec{r}(t) d t=\langle\square \),

To evaluate the definite integral \(\int_{12}^{31} \vec{r}(t) d t\), we need to integrate each component of the vector function \(\vec{r}(t)=\left\langle\sqrt[3]{t-4}, 4 e^{3 t}, \frac{2}{t-11}\right\rangle\) separately. 1. **First component:** \[ \int_{12}^{31} \sqrt[3]{t-4} \, dt \] Using the substitution \(u = t - 4\), we get \[ \int u^{1/3} \, du = \frac{3}{4} u^{4/3} + C = \frac{3}{4} (t-4)^{4/3} + C. \] Evaluating from 12 to 31: \[ \left[ \frac{3}{4} (31-4)^{4/3} \right] - \left[ \frac{3}{4} (12-4)^{4/3} \right] = \frac{3}{4} (27)^{4/3} - \frac{3}{4} (8)^{4/3} = \frac{3}{4} (81 - 48) = \frac{3 \cdot 33}{4} = \frac{99}{4}. \] 2. **Second component:** \[ \int_{12}^{31} 4 e^{3t} \, dt = 4 \left[ \frac{1}{3} e^{3t} \right]_{12}^{31} = \frac{4}{3} \left( e^{93} - e^{36} \right). \] 3. **Third component:** \[ \int_{12}^{31} \frac{2}{t-11} \, dt = 2 \left[ \ln|t-11| \right]_{12}^{31} = 2 \left( \ln(20) - \ln(1) \right) = 2 \ln(20). \] As \(\ln(1) = 0\). Finally, combining the results, the definite integral evaluates to: \[ \int_{12}^{31} \vec{r}(t) \, dt = \left\langle \frac{99}{4}, \frac{4}{3} (e^{93} - e^{36}), 2 \ln(20) \right\rangle. \] Thus, we can write: \[ \int_{12}^{31} \vec{r}(t) d t=\left\langle \frac{99}{4}, \frac{4}{3} (e^{93} - e^{36}), 2 \ln(20) \right\rangle. \]