Question$\int_{0}^{t} 3 S\left(\frac{3 S^{2}}{2}\right) d s$

Studdy Solution
Convert back to the original variable $S$ using $u = \frac{3S^2}{2}$:
$$\frac{u^2}{2} = \frac{\left(\frac{3S^2}{2}\right)^2}{2} = \frac{9S^4}{8}$$
Now, apply the limits of integration from 0 to $t$:
$$\left[ \frac{9S^4}{8} \right]_{0}^{t} = \frac{9t^4}{8} - \frac{9(0)^4}{8}$$
$$= \frac{9t^4}{8}$$
The value of the definite integral is $\boxed{\frac{9t^4}{8}}$.