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Answer 1

$\orange{ \boxed{\boxed{\begin{array}{cc}\rm \to \:given \\ \\ \sf \: velocity \: \: v = 3 + 6t + 9 {t}^{2} \\ \\ \sf \hookrightarrow \: displacement \: \: s = to \: find \\ \sf \: during \: the \: time \: interval \: t = 5s \: to \: 8s \\ \\ \\ \red{ \underline{ \sf \: solution}} \\ \\ \bf \: we \: know \: that \\ \\ \sf \: v = \frac{ds}{dt} \\ \\ \sf \implies \: ds = v \: dt \\ \\ \sf \implies \:\displaystyle \int \sf \: ds = \:\displaystyle \int \sf \:v \: dt \\ \\ \sf \implies \:s = \displaystyle \int _5^8\sf \:(3 + 6t + 9 {t}^{2} )\: dt \\ \\ \sf \implies \:s = {\huge{ [}}3t + 6 \frac{ {t}^{2} }{2} + 9 \frac{ {t}^{3} }{3}{ \huge{ ]}}_5^{8} \\ \\ \sf \implies \:s = {\huge{ [}}3t + 3 {t}^{2} + 3 {t}^{3} { \huge{ ]}}_5^{8} \\ \\ = \small{(3 \times 8 + 3 + {8}^{2} + 3 + {8}^{3} ) - (3 \times 5 + 3 \times {5}^{2} + 3 \times {5}^{3} ) }\\ \\ =1752 - 465 \\ \\ = 1287 \: cm \end{array}}}}$