Select all expressions matching the value of the definite integr int _(-3)^1e^(x)/(3)dx 3sqrt [3](e)-3e^-1 3sqrt [3](e)-(3)/(e) e^-1(3e^(4)/(3)-3) (3(e^frac (4)/(3)-1))(e)

To find the value of the definite integral $\int_{-3}^{1} e^{\frac{x}{3}} dx$, we can use the substitution method.<br /><br />Let $u = \frac{x}{3}$, then $du = \frac{1}{3}dx$, and $dx = 3du$.<br /><br />Substituting these values into the integral, we get:<br /><br />$\int_{-3}^{1} e^{\frac{x}{3}} dx = \int_{-1}^{\frac{4}{3}} e^u \cdot 3du$<br /><br />Now, we can integrate with respect to $u$:<br /><br />$\int_{-1}^{\frac{4}{3}} e^u \cdot 3du = 3 \int_{-1}^{\frac{4}{3}} e^u du$<br /><br />Using the integral of $e^u$, we have:<br /><br />$3 \int_{-1}^{\frac{4}{3}} e^u du = 3(e^u \bigg|_{-1}^{\frac{4}{3}})$<br /><br />Evaluating the definite integral, we get:<br /><br />$3(e^{\frac{4}{3}} - e^{-1})$<br /><br />Therefore, the correct answer is:<br /><br />$3\sqrt[3]{e} - 3e^{-1}$