QuestionJuly 14, 2025

Fill in the Blank 1 point Evaluate: int _(0)^13e^-2xdx=type your answer... units^2

Fill in the Blank 1 point Evaluate: int _(0)^13e^-2xdx=type your answer... units^2
Fill in the Blank 1 point Evaluate: int _(0)^13e^-2xdx=type your answer... units^2

Solution
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Answer

\frac{3}{2} (1 - e^{-2}) \, \text{units}^2 Explanation 1. Identify the integral The integral to evaluate is \int _{0}^{1}3e^{-2x}dx. 2. Apply the integration formula Use the formula for integrating e^{ax}: \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C. Here, a = -2. 3. Integrate the function Integrate: \int 3e^{-2x} \, dx = 3 \cdot \left(-\frac{1}{2} e^{-2x}\right) = -\frac{3}{2} e^{-2x}. 4. Evaluate definite integral Evaluate from 0 to 1: -\frac{3}{2} [e^{-2(1)} - e^{-2(0)}] = -\frac{3}{2} [e^{-2} - 1]. 5. Simplify the expression Simplify: -\frac{3}{2} (e^{-2} - 1) = \frac{3}{2} (1 - e^{-2}).

Explanation

1. Identify the integral<br /> The integral to evaluate is $\int _{0}^{1}3e^{-2x}dx$.<br /><br />2. Apply the integration formula<br /> Use the formula for integrating $e^{ax}$: $\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$. Here, $a = -2$.<br /><br />3. Integrate the function<br /> Integrate: $\int 3e^{-2x} \, dx = 3 \cdot \left(-\frac{1}{2} e^{-2x}\right) = -\frac{3}{2} e^{-2x}$.<br /><br />4. Evaluate definite integral<br /> Evaluate from 0 to 1: $-\frac{3}{2} [e^{-2(1)} - e^{-2(0)}] = -\frac{3}{2} [e^{-2} - 1]$.<br /><br />5. Simplify the expression<br /> Simplify: $-\frac{3}{2} (e^{-2} - 1) = \frac{3}{2} (1 - e^{-2})$.
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