To solve the integral $\int_{\ln(2)}^{e^x} \sqrt{e^x + 4} \, dx$ using substitution, let's go through the steps.

Step 1: Choose a substitution

We observe that the expression under the square root is $e^x + 4$. This suggests a substitution that will simplify the integrand. Let:

$u = e^x + 4$

Step 2: Differentiate $u$

Differentiate both sides with respect to $x$:

$\frac{du}{dx} = e^x$

This implies:

$du = e^x \, dx$

Step 3: Change the limits of integration

The original limits of integration are from $x = \ln(2)$ to $x = 0$. We need to convert these limits in terms of $u$.

For $x = \ln(2)$:

$u = e^{\ln(2)} + 4 = 2 + 4 = 6$

For $x = 0$:

$u = e^0 + 4 = 1 + 4 = 5$

So the limits of integration in terms of $u$ are from $u = 6$ to $u = 5$.

Step 4: Rewrite the integral

The integral in terms of $u$ becomes:

$\int_{6}^{5} \sqrt{u} \cdot \frac{du}{e^x}$

Since $du = e^x \, dx$, the integral simplifies to:

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

Step 5: Integrate with respect to $u$

Now, integrate $\sqrt{u}$ with respect to $u$:

$\int \sqrt{u} \, du = \int u^{1/2} \, du$

The antiderivative is:

$\frac{2}{3} u^{3/2} + C$

Step 6: Evaluate the definite integral

Now, apply the limits of integration from 6 to 5:

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

Step 7: Simplify the expression

Now compute the values:

$5^{3/2} = \sqrt{5^3} = \sqrt{125} \approx 11.1803$ $6^{3/2} = \sqrt{6^3} = \sqrt{216} \approx 14.6969$

Thus, the integral is:

$\frac{2}{3} (11.1803 - 14.6969) = \frac{2}{3} \times -3.5166 \approx -2.3444$

So, the final result is approximately:

$\boxed{-2.3444}$

Would you like further details on any of the steps?