It seems like you're looking to evaluate the integral:

$\int_0^{\ln 2} e^x \sqrt{e^x + 4} \, dx$

To solve this, we can follow these steps:

1. Substitute $u = e^x + 4$. Then, $du = e^x dx$.

2. When $x = 0$, $u = e^0 + 4 = 5$.

   When $x = \ln 2$, $u = e^{\ln 2} + 4 = 2 + 4 = 6$.

3. Now, the integral becomes:

$\int_{5}^{6} \sqrt{u} \, du$

4. Integrate $\sqrt{u}$ with respect to $u$:

$\int \sqrt{u} \, du = \frac{2}{3} u^{3/2}$

5. Apply the limits of integration from 5 to 6:

$\left[ \frac{2}{3} u^{3/2} \right]_{5}^{6} = \frac{2}{3} \left(6^{3/2} - 5^{3/2}\right)$

6. Calculate $6^{3/2}$ and $5^{3/2}$:

   • $6^{3/2} = \sqrt{6^3} = \sqrt{216}$
   • $5^{3/2} = \sqrt{5^3} = \sqrt{125}$

7. The result will be:

$\frac{2}{3} \left(\sqrt{216} - \sqrt{125}\right)$

You can simplify further if needed, but this is the evaluated form of the integral.

Would you like a more detailed explanation or to proceed with further steps?

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Here are 8 related questions:

1. How do you simplify the expression $\sqrt{216} - \sqrt{125}$?
2. Can the result of the integral be expressed in decimal form?
3. What happens if we change the limits of integration?
4. How do you integrate functions with square root terms?
5. What are the properties of definite integrals?
6. How would you approach this integral if the exponent was different?
7. Can you use numerical methods to approximate the value of this integral?
8. What is the significance of the substitution method in integration?

Tip: Always check if a substitution simplifies the integral before attempting more complex methods.