Answer
$\textbf{(a)} \ 17$
$\textbf{(b)} \ 7$
$\textbf{(c)} \ 9$
$\textbf{(d)} \ 84$
Work Step by Step
$\textbf{(a)}$ $ {\displaystyle \int_{4}^{8}}[f_{(x)} + g_{(x)}]dx = {\displaystyle \int_{4}^{8}}f_{(x)}dx + {\displaystyle \int_{4}^{8}}g_{(x)}dx = 12 + 5 = 17$
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$\textbf{(b)}$ $ {\displaystyle \int_{4}^{8}}[f_{(x)} - g_{(x)}]dx = {\displaystyle \int_{4}^{8}}f_{(x)}dx - {\displaystyle \int_{4}^{8}}g_{(x)}dx = 12 - 5 = 7$
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$\textbf{(c)} {\displaystyle \int_{4}^{8}}[2f_{(x)} - 3g_{(x)}]dx = {\displaystyle \int_{4}^{8}}2f_{(x)}dx - {\displaystyle \int_{4}^{8}}3g_{(x)}dx = \\ \quad {\displaystyle 2\int_{4}^{8}}f_{(x)}dx - {\displaystyle 3\int_{4}^{8}}g_{(x)}dx = 2\times12 - 3\times5 = 24 - 15 = 9$
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$\textbf{(d)}$ ${\displaystyle \int_{4}^{8}}7f_{(x)}dx = {\displaystyle 7\int_{4}^{8}}f_{(x)}dx = 7\times12 = 84$
