The integral given is:

$\int 9x e^{5x} \, dx$

Step 1: Solve the integral

To evaluate this integral, we will use integration by parts. The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Here, we let:

• $u = 9x$ (so that $du = 9 \, dx$),
• $dv = e^{5x} \, dx$ (so that $v = \frac{1}{5} e^{5x}$).

Using the integration by parts formula:

$\int 9x e^{5x} \, dx = 9x \cdot \frac{1}{5} e^{5x} - \int \frac{1}{5} e^{5x} \cdot 9 \, dx$

Simplify the terms:

$\int 9x e^{5x} \, dx = \frac{9x}{5} e^{5x} - \frac{9}{5} \int e^{5x} \, dx$

The integral of $e^{5x}$ is $\frac{1}{5} e^{5x}$, so:

$\int 9x e^{5x} \, dx = \frac{9x}{5} e^{5x} - \frac{9}{5} \cdot \frac{1}{5} e^{5x}$

Simplify further:

$\int 9x e^{5x} \, dx = \frac{9x}{5} e^{5x} - \frac{9}{25} e^{5x}$

Combine terms:

$\int 9x e^{5x} \, dx = e^{5x} \left( \frac{9x}{5} - \frac{9}{25} \right)$

This is the indefinite integral. Since the integral is from 0 to 0, the evaluation yields 0 because both the upper and lower bounds are the same.

Step 2: Check by Differentiating

To confirm the indefinite integral, differentiate the result:

$\frac{d}{dx} \left( e^{5x} \left( \frac{9x}{5} - \frac{9}{25} \right) \right)$

Using the product rule, let:

• First term: $u = e^{5x}$,
• Second term: $v = \frac{9x}{5} - \frac{9}{25}$.

Differentiate: $\frac{d}{dx} \left( e^{5x} \right) = 5e^{5x}, \quad \frac{d}{dx} \left( \frac{9x}{5} - \frac{9}{25} \right) = \frac{9}{5}.$

Apply the product rule:

$\frac{d}{dx} \left( e^{5x} \left( \frac{9x}{5} - \frac{9}{25} \right) \right) = e^{5x} \cdot \frac{9}{5} + 5e^{5x} \cdot \left( \frac{9x}{5} - \frac{9}{25} \right)$

Simplify:

$\frac{d}{dx} = e^{5x} \cdot \frac{9}{5} + e^{5x} \cdot \left( 9x - \frac{9}{5} \right)$

Factor $e^{5x}$:

$\frac{d}{dx} = e^{5x} \left( \frac{9}{5} + 9x - \frac{9}{5} \right)$

Combine terms:

$\frac{d}{dx} = e^{5x} \cdot 9x$

This matches the original function $9x e^{5x}$. Hence, the integral is correct.

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Final Answer

$\int_{0}^{0} 9x e^{5x} \, dx = 0$

Do you have any questions or need further explanation? 😊

Here are 5 follow-up questions:

1. Can you explain integration by parts in more detail?
2. How do we determine $u$ and $dv$ for integration by parts?
3. What is the general formula for integrating $x e^{ax}$?
4. How can we apply the limits more directly instead of integrating fully?
5. What are other methods for solving such integrals?

Tip: Always verify integrals by differentiating to ensure accuracy!