The given mathematical expression in the image is:

$\int \frac{e^{3x} + 8e^{2x} + 2e^x}{\sqrt{e^{2x} - 16}} \, dx$

To solve this integral, a substitution method may simplify the expression. Let's proceed with a step-by-step breakdown:

Step 1: Substitution

Let: $u = e^{x}$ Then: $du = e^{x} dx \quad \text{or} \quad dx = \frac{du}{u}$

This substitution transforms the integral as follows:

• $e^{2x} = u^2$
• $e^{3x} = u^3$
• $e^x = u$

So, the integral becomes: $\int \frac{u^3 + 8u^2 + 2u}{\sqrt{u^2 - 16}} \cdot \frac{du}{u}$

Step 2: Simplification

This can now be simplified further: $\int \frac{u^2 + 8u + 2}{\sqrt{u^2 - 16}} \, du$

Now, the problem reduces to integrating this expression. This involves handling the algebraic expression inside the integral and potentially splitting it into simpler terms that can be tackled individually, often using trigonometric substitution for the square root.

Shall I continue and solve the integral fully using trigonometric substitution, or would you like a deeper explanation of each step so far?

Follow-up questions:

1. Would you like more clarification on how the substitution was chosen?
2. Should I proceed with the trigonometric substitution for the square root term?
3. Are you familiar with solving integrals of this type using trigonometric identities?
4. Would you like to see how this integral is split into simpler fractions?
5. Do you want to explore an alternative method, such as integration by parts, instead?

Tip: Substitution is often the most effective method for simplifying exponential functions inside integrals.