To evaluate the indefinite integral $\int (9t + 8)^{2.2} \, dt$, we'll use the following approach:

Step 1: Apply a substitution.

Let’s perform a substitution to simplify the integral. Set:

$u = 9t + 8$

Then, differentiate both sides with respect to $t$:

$du = 9 \, dt \quad \Rightarrow \quad dt = \frac{du}{9}$

Step 2: Rewrite the integral.

Substitute $u = 9t + 8$ and $dt = \frac{du}{9}$ into the integral:

$\int (9t + 8)^{2.2} \, dt = \int u^{2.2} \cdot \frac{du}{9}$

This simplifies to:

$\frac{1}{9} \int u^{2.2} \, du$

Step 3: Integrate.

Now, integrate $u^{2.2}$:

$\int u^{2.2} \, du = \frac{u^{2.2 + 1}}{2.2 + 1} = \frac{u^{3.2}}{3.2}$

Step 4: Substitute back $u = 9t + 8$.

Now, substitute back $u = 9t + 8$ to return to the original variable:

$\frac{1}{9} \cdot \frac{(9t + 8)^{3.2}}{3.2}$

Final Answer:

$\frac{(9t + 8)^{3.2}}{9 \cdot 3.2} + C$ $= \frac{(9t + 8)^{3.2}}{28.8} + C$

Where $C$ is the constant of integration.

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Would you like more details on any of the steps? Let me know if you have any questions!

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Here are some related questions for further exploration:

1. How does the substitution method work in general for solving integrals?
2. What is the rule for integrating a power of a linear expression?
3. Can this method be extended to integrals of trigonometric functions?
4. What happens if the exponent were a negative number, e.g., $(9t + 8)^{-2.2}$?
5. How do we approach the definite integral of a similar expression?

Tip: When dealing with powers of a linear function, substitution is often a helpful technique, as it simplifies the integration process significantly.