find the antiderivative of the given derivative.\n\\( \\frac { d s } { d t } = 14 t \\left( 4 t ^ { 2 } - 5…

Explanation:

Step1: Use substitution

Let (u = 4t^{2}-5), then (du=8tdt), and (tdt=\frac{1}{8}du). The integral (\int14t(4t^{2} - 5)^{3}dt) becomes (\int14u^{3}\times\frac{1}{8}du).

Step2: Simplify the integral

(\int14u^{3}\times\frac{1}{8}du=\frac{14}{8}\int u^{3}du=\frac{7}{4}\int u^{3}du).

Step3: Integrate using the power rule

Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), for (n = 3), (\frac{7}{4}\int u^{3}du=\frac{7}{4}\times\frac{u^{4}}{4}+C=\frac{7u^{4}}{16}+C).

Step4: Substitute back (u = 4t^{2}-5)

Substituting (u = 4t^{2}-5) back, we get (s=\frac{7(4t^{2}-5)^{4}}{16}+C).

Answer:

(s=\frac{7(4t^{2}-5)^{4}}{16}+C)