We are tasked with solving the definite integral:

$I = \int_0^{\ln 6} 3e^x \sqrt{e^x + 3} \, dx$

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Step 1: Substitution

To simplify the integral, we perform a substitution. Let:

$u = e^x + 3$

Now, differentiate $u$ with respect to $x$:

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

The boundaries will also change based on $u = e^x + 3$:

• When $x = 0$, $u = e^0 + 3 = 1 + 3 = 4$,
• When $x = \ln 6$, $u = e^{\ln 6} + 3 = 6 + 3 = 9$.

Thus, the new limits are $u = 4$ to $u = 9$.

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Step 2: Expressing the integral

From the substitution $u = e^x + 3$, it follows that $e^x = u - 3$. The integral becomes:

$I = \int_4^9 3e^x \sqrt{e^x + 3} \, dx$

Substitute $e^x = u - 3$ and $dx = \frac{du}{e^x} = \frac{du}{u - 3}$. Also, $\sqrt{e^x + 3} = \sqrt{u}$. Therefore:

$I = \int_4^9 3(u - 3) \sqrt{u} \frac{du}{u - 3}$

Simplify the expression. Notice that the $u - 3$ terms cancel out:

$I = \int_4^9 3 \sqrt{u} \, du$

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Step 3: Solve the integral

The integral of $\sqrt{u}$ can be written as:

$\sqrt{u} = u^{\frac{1}{2}} \quad \implies \quad \int u^{\frac{1}{2}} \, du = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} u^{\frac{3}{2}}$

Thus:

$\int_4^9 3 \sqrt{u} \, du = 3 \cdot \frac{2}{3} \left[ u^{\frac{3}{2}} \right]_4^9$

Simplify the constants:

$\int_4^9 3 \sqrt{u} \, du = 2 \left[ u^{\frac{3}{2}} \right]_4^9$

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Step 4: Evaluate the limits

Now evaluate $u^{\frac{3}{2}}$ at $u = 9$ and $u = 4$:

• At $u = 9$: $9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}} = 3^3 = 27$,
• At $u = 4$: $4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8$.

Subtract the values:

$\left[ u^{\frac{3}{2}} \right]_4^9 = 27 - 8 = 19$

Thus:

$I = 2 \cdot 19 = 38$

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Final Answer:

The value of the definite integral is:

$\boxed{38}$

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Do you need further clarification or a detailed breakdown of any step?

Here are 5 related questions to explore further:

1. How to evaluate definite integrals using substitution?
2. What is the general formula for integrating $\sqrt{x}$?
3. How do you handle integrals with exponential functions and square roots?
4. How do you change limits of integration under substitution?
5. What are the properties of definite integrals and their applications?

Tip: When using substitution, always remember to adjust the limits of integration to match the new variable.