Key Concepts and Formulas

• Indefinite Integration: The process of finding the antiderivative of a function.
• Substitution Method: A technique to simplify integrals by substituting a part of the integrand with a new variable.
• Integral Formula: $\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln |x + \sqrt{x^2 + a^2}| + C$

Step-by-Step Solution

Step 1: Set up the integral and perform the initial substitution. We are given the integral $I(x) = \int \sqrt{\frac{x+7}{x}} \, dx$. To simplify, we substitute $x = t^2$. This implies $dx = 2t \, dt$. The integral becomes: $I(x) = \int \sqrt{\frac{t^2+7}{t^2}} \cdot 2t \, dt = \int \frac{\sqrt{t^2+7}}{t} \cdot 2t \, dt = 2 \int \sqrt{t^2+7} \, dt$ Reasoning: The substitution helps to remove the fraction inside the square root, making the integral easier to handle.

Step 2: Apply the standard integral formula. We use the formula $\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln |x + \sqrt{x^2 + a^2}| + C$. In our case, $x=t$ and $a^2 = 7$, so $a = \sqrt{7}$. $I(t) = 2 \left[ \frac{t}{2} \sqrt{t^2+7} + \frac{7}{2} \ln |t + \sqrt{t^2+7}| \right] + C$ $I(t) = t \sqrt{t^2+7} + 7 \ln |t + \sqrt{t^2+7}| + C$ Reasoning: This step directly applies a known integral formula to solve the integral in terms of $t$.

Step 3: Substitute back to express the integral in terms of $x$. Since $x = t^2$, we have $t = \sqrt{x}$. Substituting back into the expression for $I(t)$, we get: $I(x) = \sqrt{x} \sqrt{x+7} + 7 \ln |\sqrt{x} + \sqrt{x+7}| + C$ Reasoning: This step reverses the initial substitution to express the result in terms of the original variable $x$.

Step 4: Use the given condition $I(9) = 12 + 7 \ln 7$ to find the constant $C$. We are given that $I(9) = 12 + 7 \ln 7$. Substituting $x = 9$ into the expression for $I(x)$, we have: $I(9) = \sqrt{9} \sqrt{9+7} + 7 \ln |\sqrt{9} + \sqrt{9+7}| + C$ $12 + 7 \ln 7 = 3 \sqrt{16} + 7 \ln |3 + \sqrt{16}| + C$ $12 + 7 \ln 7 = 3 \cdot 4 + 7 \ln |3 + 4| + C$ $12 + 7 \ln 7 = 12 + 7 \ln 7 + C$ Therefore, $C = 0$. Reasoning: Using the given initial condition, we can solve for the constant of integration.

Step 5: Calculate $I(1)$. Now that we know $C = 0$, we have $I(x) = \sqrt{x} \sqrt{x+7} + 7 \ln |\sqrt{x} + \sqrt{x+7}|$. We need to find $I(1)$: $I(1) = \sqrt{1} \sqrt{1+7} + 7 \ln |\sqrt{1} + \sqrt{1+7}|$ $I(1) = 1 \cdot \sqrt{8} + 7 \ln |1 + \sqrt{8}|$ $I(1) = \sqrt{8} + 7 \ln (1 + 2\sqrt{2})$ Reasoning: We substitute $x=1$ into the expression for $I(x)$ to find the value of the integral at $x=1$.

Step 6: Determine $\alpha$ and calculate $\alpha^4$. We are given that $I(1) = \alpha + 7 \ln (1 + 2\sqrt{2})$. Comparing this with our calculated value of $I(1) = \sqrt{8} + 7 \ln (1 + 2\sqrt{2})$, we find that $\alpha = \sqrt{8}$. Therefore, $\alpha^4 = (\sqrt{8})^4 = (8^{1/2})^4 = 8^2 = 64$. Reasoning: By comparing the given form of $I(1)$ with our calculated value, we can isolate the value of $\alpha$ and then calculate $\alpha^4$.

Common Mistakes & Tips

• Forgetting the Constant of Integration: Always remember to add the constant of integration, $C$, when evaluating indefinite integrals.
• Incorrect Substitution: Ensure that the substitution and the differential are correctly calculated.
• Simplification Errors: Double-check your algebraic manipulations to avoid errors.

Summary We solved the indefinite integral $I(x) = \int \sqrt{\frac{x+7}{x}} \, dx$ using substitution and a standard integral formula. We then used the given condition $I(9) = 12 + 7 \ln 7$ to find the constant of integration, $C=0$. Next, we calculated $I(1)$ and compared it with the given form $I(1) = \alpha + 7 \ln (1 + 2\sqrt{2})$ to find $\alpha = \sqrt{8}$. Finally, we calculated $\alpha^4 = (\sqrt{8})^4 = 64$.

Final Answer The final answer is \boxed{64}.