find the antiderivative of the given derivative.\n\\( \\frac { d s } { d t } = 8 t ( 5 t ^ { 2 } - 3 ) ^ { 3…

Explanation:

Step1: Use substitution method

Let (u = 5t^{2}-3), then (du=10t\space dt), and (t\space dt=\frac{1}{10}du). The integral (\int8t(5t^{2} - 3)^{3}dt) can be rewritten as (\int8\times\frac{1}{10}u^{3}du=\frac{4}{5}\int u^{3}du).

Step2: Integrate (u) - term

According to the power - rule of integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), for (\int u^{3}du), we have (\frac{u^{4}}{4}+C).

Step3: Substitute back (u)

Substitute (u = 5t^{2}-3) into (\frac{4}{5}\times\frac{u^{4}}{4}+C). We get (\frac{(5t^{2}-3)^{4}}{5}+C).

Answer:

(\frac{(5t^{2}-3)^{4}}{5}+C)