Question

If \[ \int e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) \, dx = f(x) + c \quad \text{and} \quad f(0) = 1 \] find \( f\left( \frac{1}{2} \right) \):

• A. \( 2 + \sqrt{3e} \)
• B. \( 2 - \sqrt{3e} \)
• C. \( 2 + \sqrt{e} \)
• D. \( 2 - \sqrt{e} \)

Hint:

When solving integrals involving complicated expressions, simplify the problem step by step and use the given initial conditions to find the constants.

Answer 1

The Correct Option is B

Step 1: Differentiate the given equation.
We are given the equation: \[ \int e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) \, dx = f(x) + c \] To find \( f(x) \), differentiate both sides with respect to \( x \). This gives: \[ e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) = f'(x) \]
Step 2: Solve for \( f(x) \).
We now have: \[ f'(x) = e^x \left( \frac{x^2 - 2}{\sqrt{1 + x(1 - x)^{3/2}}} \right) \] Integrate both sides of the equation to get \( f(x) \).
Step 3: Use the initial condition \( f(0) = 1 \).
Use the initial condition \( f(0) = 1 \) to determine the constant of integration \( c \). After solving, we find: \[ f(x) = 2 - \sqrt{3e} \] Final Answer: \[ \boxed{2 - \sqrt{3e}} \]