To integrate the expression

$\int \frac{dt}{t^{4/3}}$

we can rewrite the integrand in a simpler form. Notice that:

$\frac{1}{t^{4/3}} = t^{-4/3}$

So the integral becomes:

$\int t^{-4/3} \, dt$

Step 1: Apply the Power Rule

Using the power rule for integration, which states that $\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$ (where $n \neq -1$), we can integrate $t^{-4/3}$.

Here, $n = -\frac{4}{3}$, so:

$\int t^{-4/3} \, dt = \frac{t^{-4/3 + 1}}{-4/3 + 1} + C$

Step 2: Simplify the Exponent

Calculate $-\frac{4}{3} + 1$:

$-\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}$

Thus, we have:

$\int t^{-4/3} \, dt = \frac{t^{-1/3}}{-1/3} + C$

Step 3: Simplify the Fraction

Dividing by $-\frac{1}{3}$ is equivalent to multiplying by $-3$:

$= -3t^{-1/3} + C$

Step 4: Rewrite in Terms of Positive Exponents

Finally, we can write $t^{-1/3}$ as $\frac{1}{t^{1/3}}$:

$\int \frac{dt}{t^{4/3}} = -\frac{3}{t^{1/3}} + C$

Final Answer

$\int \frac{dt}{t^{4/3}} = -\frac{3}{t^{1/3}} + C$

Would you like a detailed explanation of each step, or do you have any questions?

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Here are 5 related questions to deepen understanding:

1. How would you evaluate $\int \frac{dt}{t^{5/3}}$?
2. What is the general formula for integrating expressions of the form $\int t^n \, dt$?
3. How does the result change if we integrate $\int \frac{dt}{t^{2}}$ instead?
4. What is the power rule, and why does it not apply to $\int \frac{1}{t} \, dt$?
5. How can we apply similar techniques to evaluate $\int \frac{1}{t^{a}} \, dt$ for any constant $a$?

Tip: When working with fractional exponents, rewrite terms to apply integration rules more easily.