Question

find all functions f(t) that satisfy the given condition
f(t)=t^{\frac{7}{3}}
f(t)=\square

Explanation:

Step1: Apply the power rule for integration

The power rule for integration is $\int t^n dt=\frac{t^{n + 1}}{n+1}+C$, where $n
eq - 1$ and $C$ is the constant of integration.
For $n=\frac{7}{3}$, we have:
$$\int t^{\frac{7}{3}}dt=\frac{t^{\frac{7}{3}+1}}{\frac{7}{3}+1}+C$$

Step2: Simplify the exponent and the denominator

First, simplify the exponent: $\frac{7}{3}+1=\frac{7 + 3}{3}=\frac{10}{3}$.
Then, simplify the denominator: $\frac{7}{3}+1=\frac{10}{3}$.
So, $\frac{t^{\frac{7}{3}+1}}{\frac{7}{3}+1}+C=\frac{t^{\frac{10}{3}}}{\frac{10}{3}}+C$.
Using the rule $\frac{a}{b/c}=\frac{ac}{b}$, we get $\frac{3t^{\frac{10}{3}}}{10}+C$.

Answer:

$f(t)=\frac{3}{10}t^{\frac{10}{3}}+C$