10. Evaluate the indefinite integral. (Use $$ C $$ for the constant of integration.) $$ \int e^{x} \sqrt{5+e^{x}} d x $$

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To evaluate the indefinite integral

$$
\int e^{x} \sqrt{5+e^{x}} \, dx,
$$

we can use a substitution method. Let's define a new variable:

$$
u = 5 + e^{x}.
$$

Then, we differentiate $$u$$ with respect to $$x$$:

$$
\frac{du}{dx} = e^{x} \quad \Rightarrow \quad du = e^{x} \, dx.
$$

Now, we can rewrite the integral in terms of $$u$$:

$$
\int e^{x} \sqrt{5+e^{x}} \, dx = \int \sqrt{u} \, du.
$$

Next, we can evaluate the integral of $$\sqrt{u}$$:

$$
\int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} + C = \frac{2}{3} u^{3/2} + C.
$$

Now, we substitute back $$u = 5 + e^{x}$$:

$$
\frac{2}{3} (5 + e^{x})^{3/2} + C.
$$

Thus, the final result for the indefinite integral is:

$$
\int e^{x} \sqrt{5+e^{x}} \, dx = \frac{2}{3} (5 + e^{x})^{3/2} + C.
$$
The indefinite integral evaluates to $$ \frac{2}{3} (5 + e^{x})^{3/2} + C $$.

10. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int e^{x} \sqrt{5+e^{x}} d x \]

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