Question

If \[ \int(e^{2x}+2e^x)\sqrt{e^{2x}-4e^x+5}\,dx = \frac13[f(x)]^{3/2} + 4\left[ \frac{e^x-2}{2}\sqrt{f(x)} + \frac12g(x) \right]+c \] then \(f(0)=\)

• A. \(2\)
• B. \(0\)
• C. \(1\)
• D. \(3\)

Hint:

In integration pattern questions, first identify repeated expression hidden inside the final answer structure.

Answer 1

The Correct Option is A

Step 1: Read the structure of the answer.
The right side is built around $\sqrt{f(x)}$ and $[f(x)]^{3/2}$, which means $f(x)$ is exactly the quantity sitting under the square root in the integrand.
Step 2: Identify the radicand.
Inside the integral we see $\sqrt{e^{2x}-4e^x+5}$, so the natural choice is $f(x)=e^{2x}-4e^x+5$.
Step 3: Sanity-check with the derivative.
$f'(x)=2e^{2x}-4e^x=2e^x(e^x-2)$, which matches the factor $\dfrac{e^x-2}{2}$ appearing in the answer, confirming our choice of $f$.
Step 4: Substitute $x=0$ into $f$.
$f(0)=e^{0}\cdot e^{0}-4e^{0}+5=1-4+5$.
Step 5: Simplify.
$1-4+5=2$.
Step 6: State the result.
Hence $f(0)=2$, matching option (1).
\[ \boxed{2} \]