Question

If \(f^'(x) = \frac{(\sqrt{x}+1)e^{\sqrt{x}}}{\sqrt{x}}\) and \(f(0) = e\) then \(f(1) = \ldots \ldots \ldots \ldots\)

• A. \(e\)
• B. \(2e\)
• C. \(3e\)
• D. \(4e\)

Hint:

Split \(f'\) and integrate each term using \(t=\sqrt x\).

Answer 1

The Correct Option is C

Step 1: Guess the antiderivative
Try $F=2\sqrt x\,e^{\sqrt x}$. Then $F'=\dfrac{e^{\sqrt x}}{\sqrt x}+2\sqrt x\,e^{\sqrt x}\cdot\dfrac{1}{2\sqrt x}=\dfrac{e^{\sqrt x}}{\sqrt x}+e^{\sqrt x}$, which equals $f'$.

Step 2: Use the condition
$f=2\sqrt xe^{\sqrt x}+C$ and $f(0)=e$ gives $C=e$. So $f(1)=2e+e=3e$, option (C).

Final Answer:
The value is $3e$, option (C). \[ \boxed{3e} \]