To solve the integral $\int_0^9 2.9e^{0.05x} \, dx$ using the U-substitution method, follow these steps:

Step 1: Identify the substitution

Let: $u = 0.05x$ Then, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 0.05 \quad \Rightarrow \quad dx = \frac{du}{0.05}$

Step 2: Rewrite the integral

Now substitute $u = 0.05x$ and $dx = \frac{du}{0.05}$: $\int_0^9 2.9e^{0.05x} \, dx = \int_0^{0.05 \cdot 9} 2.9e^u \cdot \frac{du}{0.05}$ Simplify the constants: $\int_0^{0.45} \frac{2.9}{0.05} e^u \, du = \int_0^{0.45} 58e^u \, du$

Step 3: Integrate

The integral of $e^u$ is $e^u$, so: $58 \int_0^{0.45} e^u \, du = 58 \left[ e^u \right]_0^{0.45}$ Evaluating at the limits: $58 \left( e^{0.45} - e^0 \right) = 58 \left( e^{0.45} - 1 \right)$

Step 4: Final answer

Now, calculate the values of $e^{0.45}$: $e^{0.45} \approx 1.5683$ Thus: $58 \left( 1.5683 - 1 \right) = 58 \times 0.5683 \approx 32.961$

So, the value of the integral is approximately: $\boxed{32.96}$

Would you like to see more details or clarifications on any of the steps?

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Here are 5 questions related to this process:

1. How does substitution work in integrals, and why is it useful?
2. What is the significance of the exponential function $e^x$ in calculus?
3. Can you explain why we change the limits of integration after substitution?
4. How would this method change if the exponent were more complex, such as $e^{x^2}$?
5. How do you compute the value of $e^x$ manually or using a calculator?

Tip: Always simplify constants outside the integral before proceeding with the actual integration to reduce complexity!