Find the antiderivative of the given derivative. \[ \frac{d s}{d t}=12 t\left(4 t^{2}-7\right)^{3} \] \[ s=\square \]

Step 1 ：Given the derivative \(\frac{d s}{d t}=12 t\left(4 t^{2}-7\right)^{3}\)

Step 2 ：Recognize this as a case of the chain rule for derivatives, where the outer function is \((u)^3\) and the inner function is \(4t^2 - 7\), with its derivative being \(8t\)

Step 3 ：Use the method of substitution. Let's set \(u = 4t^2 - 7\). Then, \(du = 8t dt\)

Step 4 ：Rewrite 12 as 3*4, and then the integral becomes \(\int 3*4t\left(4 t^{2}-7\right)^{3} dt\)

Step 5 ：Substitute \(u\) and \(du\) into the integral: \(\int 3u^3 * (1/4) du\)

Step 6 ：Solve this integral to get \(s = (3/4) * (1/4)u^4 + C\)

Step 7 ：Substitute back for \(u\) to get \(s = (3/16)(4t^2 - 7)^4 + C\)

Step 8 ：So, the antiderivative of the given derivative is \(\boxed{s = (3/16)(4t^2 - 7)^4 + C}\)