We are asked to integrate \( e^{ax} \cos(ax) \sin(ax) \, dx \).

To approach this, let's first simplify the expression. Notice that:

\[
\cos(ax) \sin(ax) = \frac{1}{2} \sin(2ax)
\]

Thus, the integral becomes:

\[
\int e^{ax} \cos(ax) \sin(ax) \, dx = \frac{1}{2} \int e^{ax} \sin(2ax) \, dx
\]

Now, we focus on solving:

\[
\frac{1}{2} \int e^{ax} \sin(2ax) \, dx
\]

This integral can be solved using the method of integration by parts. Let’s set it up using the standard formula for integration by parts:

\[
\int u \, dv = uv - \int v \, du
\]

Here, we choose:

- \( u = e^{ax} \), so \( du = a e^{ax} \, dx \)
- \( dv = \sin(2ax) \, dx \), so \( v = -\frac{1}{2a} \cos(2ax) \)

Applying integration by parts:

\[
\int e^{ax} \sin(2ax) \, dx = -\frac{1}{2a} e^{ax} \cos(2ax) + \frac{1}{2a} \int e^{ax} \cos(2ax) \, dx
\]

Now, for the remaining integral \( \int e^{ax} \cos(2ax) \, dx \), we apply integration by parts again, similarly as above. After solving both parts, you should get the final answer as a combination of exponential and trigonometric terms.

The final solution will be of the form:

\[
\frac{1}{10a} e^{ax} \sin(2ax) + \frac{1}{10} e^{ax} \cos(2ax) + C
\]

Would you like a detailed step-by-step breakdown of this solution?

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**Related Questions:**

1. How do you apply integration by parts to more complex trigonometric integrals?
2. What is the general approach for integrals of the form \( e^{mx} \sin(nx) \)?
3. Can you demonstrate how to solve a similar problem involving \( e^{ax} \cos(2ax) \)?
4. How would this process differ if we had a different combination of trigonometric functions?
5. What alternative methods can be used for solving such integrals?

**Tip:** Always simplify trigonometric expressions using identities before attempting integration.

Explore the step-by-step solution for integrating e^ax * cos(ax) * sin(ax) using the integration by parts method. Learn how to simplify trigonometric expressions and apply key formulas such as the trigonometric identity for cos(ax)sin(ax) and integration by parts. Ideal for calculus students at Grades 11-12 or university level.

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