Key Concepts and Formulas

• Indefinite Integration: Finding a function whose derivative is a given function. The result includes an arbitrary constant of integration, denoted by $C$.
• Substitution Method: A technique to simplify integrals by substituting a part of the integrand with a new variable. If $\int f(g(x))g'(x) \, dx$ is the integral, substitute $u = g(x)$, then $du = g'(x) \, dx$, and the integral becomes $\int f(u) \, du$.

Step-by-Step Solution

Step 1: Evaluate the indefinite integral using substitution.

We are given the integral $\int x^3 \sqrt{3-x^2} \, dx$. We will use the substitution method to simplify this integral. Let $t^2 = 3 - x^2$. This substitution is chosen because it simplifies the square root term.

Differentiating both sides with respect to $x$, we get $2t \, dt = -2x \, dx$, which simplifies to $x \, dx = -t \, dt$. Also, from $t^2 = 3 - x^2$, we have $x^2 = 3 - t^2$. Thus, $x^3 \, dx = x^2 \cdot x \, dx = (3-t^2)(-t \, dt) = (t^3 - 3t) \, dt$.

Therefore,

$$\int x^3 \sqrt{3-x^2} \, dx = \int (3-t^2) \cdot t \cdot (-t \, dt) = \int (t^4 - 3t^2) \, dt$$

Step 2: Integrate the simplified expression.

Now, we integrate the expression with respect to $t$:

$$\int (t^4 - 3t^2) \, dt = \frac{t^5}{5} - t^3 + C$$

Step 3: Substitute back to express the result in terms of x.

We need to substitute $t = \sqrt{3-x^2}$ back into the expression:

$$f(x) = \frac{(3-x^2)^{5/2}}{5} - (3-x^2)^{3/2} + C$$

Step 4: Determine the constant of integration using the given condition.

We are given that $5f(\sqrt{2}) = -4$, which means $f(\sqrt{2}) = -\frac{4}{5}$. Substitute $x = \sqrt{2}$ into the expression for $f(x)$:

$$f(\sqrt{2}) = \frac{(3-(\sqrt{2})^2)^{5/2}}{5} - (3-(\sqrt{2})^2)^{3/2} + C = \frac{(3-2)^{5/2}}{5} - (3-2)^{3/2} + C = \frac{1}{5} - 1 + C$$

Since $f(\sqrt{2}) = -\frac{4}{5}$, we have

$$-\frac{4}{5} = \frac{1}{5} - 1 + C$$

Solving for $C$, we get

$$C = -\frac{4}{5} - \frac{1}{5} + 1 = -\frac{5}{5} + 1 = -1 + 1 = 0$$

Thus, $C = 0$.

Step 5: Write the expression for f(x) with the determined constant.

Now we have the complete expression for $f(x)$:

$$f(x) = \frac{(3-x^2)^{5/2}}{5} - (3-x^2)^{3/2}$$

Step 6: Calculate f(1).

We want to find $f(1)$. Substitute $x = 1$ into the expression for $f(x)$:

$$f(1) = \frac{(3-1^2)^{5/2}}{5} - (3-1^2)^{3/2} = \frac{(2)^{5/2}}{5} - (2)^{3/2} = \frac{2^{2} \cdot 2^{1/2}}{5} - 2^{1/2} \cdot 2 = \frac{4\sqrt{2}}{5} - 2\sqrt{2}$$ $$f(1) = \sqrt{2} \left( \frac{4}{5} - 2 \right) = \sqrt{2} \left( \frac{4 - 10}{5} \right) = \sqrt{2} \left( -\frac{6}{5} \right) = -\frac{6\sqrt{2}}{5}$$

Common Mistakes & Tips

• Sign Errors: Pay close attention to the signs when performing the substitution and integrating. A common mistake is to miss the negative sign when differentiating $3-x^2$.
• Back-Substitution: Remember to substitute back to the original variable after integrating.
• Constant of Integration: Don't forget to add the constant of integration, $C$, after performing the indefinite integral. Use the given condition to determine the value of $C$.

Summary

We first evaluated the indefinite integral $\int x^3 \sqrt{3-x^2} \, dx$ using the substitution method. We then used the given condition $5f(\sqrt{2}) = -4$ to find the constant of integration. Finally, we substituted $x = 1$ into the expression for $f(x)$ to find $f(1)$, which is $-\frac{6\sqrt{2}}{5}$.

Final Answer

The final answer is \boxed{-\frac{6 \sqrt{2}}{5}}, which corresponds to option (A).