The problem in the image asks to "Demonstrate how to find the integral" of the expression:

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

Solution: Integration by Parts

This integral can be solved using integration by parts. The formula for integration by parts is:

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

1. Choose $u$ and $dv$:

   • Let $u = x$, which implies $du = dx$.
   • Let $dv = 9e^{5x} \, dx$, which implies that we need to find $v$ by integrating $dv$.

2. Find $v$: $v = \int 9e^{5x} \, dx = \frac{9}{5} e^{5x}$ (since the integral of $e^{ax}$ is $\frac{1}{a} e^{ax}$).

3. Apply the integration by parts formula: Using the formula $\int u \, dv = uv - \int v \, du$: $\int 9xe^{5x} \, dx = x \cdot \frac{9}{5} e^{5x} - \int \frac{9}{5} e^{5x} \, dx$

4. Simplify the remaining integral: $\int \frac{9}{5} e^{5x} \, dx = \frac{9}{25} e^{5x}$

5. Final result: Substituting this back, we get: $\int 9xe^{5x} \, dx = \frac{9}{5} x e^{5x} - \frac{9}{25} e^{5x} + C$ where $C$ is the constant of integration.

Thus, the integral of $9xe^{5x} \, dx$ is:

$\boxed{\frac{9}{5} x e^{5x} - \frac{9}{25} e^{5x} + C}$

Would you like further explanation or have any specific questions on the process?

Related Questions:

1. How does integration by parts work for other types of functions like logarithmic or trigonometric ones?
2. What are the conditions under which integration by parts is the best technique to use?
3. How can we verify the result of an integral using differentiation?
4. Can we generalize this process for integrals involving higher powers of $x$?
5. What is the impact of different coefficients in the exponential term (e.g., $e^{ax}$) on the integration?

Tip:

When selecting $u$ in integration by parts, it's