The integral given in the image is:

$\int \frac{e^{2x} - e^{4x}}{e^x - e^{-x}} \, dx = 9$

Solution:

1. Simplify the numerator and denominator:

   Let’s rewrite the numerator and denominator:

   • The numerator is $e^{2x} - e^{4x}$.
   • The denominator is $e^x - e^{-x}$, which can also be written as $2\sinh(x)$.

   So the integral becomes:

   = \int \frac{e^{2x}(1 - e^{2x})}{2\sinh(x)} \, dx$$

2. Simplify further using substitutions:

   Set $u = e^x$, which implies $du = e^x dx$. Then:

   • $e^{2x} = u^2$,
   • $e^{4x} = u^4$,
   • $e^x - e^{-x} = u - \frac{1}{u}$.

   Substitute into the integral:

   = \int \frac{u^2(1 - u^2)}{\frac{u^2 - 1}{u}} \frac{1}{2} \, du$$

3. Simplify the fraction:

   Combine terms:

   = u^3 \cdot \frac{1 - u^2}{u^2 - 1}.$$ Since $$1 - u^2 = -(u^2 - 1)$$, this simplifies to: $$-u^3.$$ The integral becomes: $$\int -\frac{u^3}{2} \, du.$$

4. Integrate:

   Compute the integral:

   = -\frac{1}{2} \cdot \frac{u^4}{4} + C = -\frac{u^4}{8} + C.$$

5. Back-substitute $u = e^x$:

   Recall $u = e^x$, so:

   = -\frac{e^{4x}}{8} + C.$$ This matches the integral and the value being equated to $$9$$ helps find bounds or constants.

Would you like me to simplify further or clarify any step? Let me know if you'd like me to check limits or adjust this!

Related Questions:

1. How can substitutions simplify complex exponential integrals further?
2. What role does hyperbolic functions play in exponential equations?
3. Can integration be approached differently using series expansions?
4. What steps ensure accuracy in back-substitutions for indefinite integrals?
5. How to verify constants in definite integrals mathematically?

Tip: Always double-check substitutions to simplify the evaluation process for exponential integrals.