evaluate the definite integral below. \n\\(\\int_{3}^{5}(6 + 7e^{0.5x})dx\\)

Explanation:

Step1: Use integral rules

We know that $\int (a + b)dx=\int a dx+\int b dx$. So, $\int_{3}^{5}(6 + 7e^{0.5x})dx=\int_{3}^{5}6dx+\int_{3}^{5}7e^{0.5x}dx$.

Step2: Integrate the first - part

The integral of a constant $k$ is $kx + C$. So, $\int_{3}^{5}6dx=6x\big|_{3}^{5}=6\times(5 - 3)=12$.

Step3: Integrate the second - part

Let $u = 0.5x$, then $du=0.5dx$ and $dx = 2du$. When $x = 3$, $u = 1.5$; when $x = 5$, $u = 2.5$. So, $\int_{3}^{5}7e^{0.5x}dx=7\times2\int_{1.5}^{2.5}e^{u}du = 14e^{u}\big|_{1.5}^{2.5}=14(e^{2.5}-e^{1.5})$.

Step4: Combine the results

$\int_{3}^{5}(6 + 7e^{0.5x})dx=12+14(e^{2.5}-e^{1.5})$. Calculate $e^{2.5}\approx12.18249$, $e^{1.5}\approx4.48169$. $12+14(12.18249 - 4.48169)=12+14\times7.7008=12 + 107.8112=119.8112\approx119.81$.

Answer:

$119.81$