Math / Calculus

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

Studdy Solution

STEP 1

1. We are given a definite integral to evaluate.
2. The integrand is $3S\left(\frac{3S^2}{2}\right)$ , where $S$ is the variable of integration.
3. The limits of integration are from 0 to $t$ .

STEP 2

1. Simplify the integrand if possible.
2. Determine if substitution is needed.
3. Evaluate the indefinite integral.
4. Apply the limits of integration to find the definite integral.

STEP 3

The integrand is $3S\left(\frac{3S^2}{2}\right)$ . At first glance, it appears that the function inside the integral might be simplified or prepared for substitution. However, without additional context or simplification, we will proceed to consider substitution directly.

STEP 4

Consider using substitution to simplify the integral. Let:
$u = \frac{3S^2}{2}$
Then, differentiate $u$ with respect to $S$ :
$\frac{du}{dS} = 3S$
This implies:
$du = 3S \, dS$
This substitution simplifies the integral significantly.

STEP 5

Substitute $u = \frac{3S^2}{2}$ and $du = 3S \, dS$ into the integral:
$\int 3S\left(\frac{3S^2}{2}\right) \, dS = \int u \, du$
Now, integrate with respect to $u$ :
$\int u \, du = \frac{u^2}{2} + C$

STEP 6

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 $\boxed{\frac{9t^4}{8}}$ .

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