Question

If \[ \int\frac{e^x+3}{\sqrt{e^x+4}}\,dx = 2\sqrt{e^x+4} +\frac32 \log\left| \frac{f(x)-2}{f(x)+2} \right| +c \] and \[ f(0)=\sqrt5, \] then \[ f(\log_e5)= \]

• A. \(\sqrt5\)
• B. \(2\)
• C. \(3\)
• D. \(\sqrt7\)

Hint:

Remember, \[ \boxed{ e^{\log_e a}=a. } \] Also, compare the given integral with the obtained result to identify the unknown function.

Answer 1

The Correct Option is C

Step 1: Identify the function \(f(x)\). The logarithmic term obtained while integrating \[ \int\frac{e^x+3}{\sqrt{e^x+4}}\,dx \] is \[ \frac32 \log \left| \frac{\sqrt{e^x+4}-2}{\sqrt{e^x+4}+2} \right|. \] Comparing with \[ \frac32 \log \left| \frac{f(x)-2}{f(x)+2} \right|, \] we obtain \[ \boxed{ f(x)=\sqrt{e^x+4}. } \] This also satisfies \[ f(0)=\sqrt{1+4}=\sqrt5. \]

Step 2: Evaluate \(f(\log_e5)\). Since \[ e^{\log_e5}=5, \] we get \[ f(\log_e5) = \sqrt{5+4} = \sqrt9 = 3. \] Hence, \[ \boxed{f(\log_e5)=3.} \] Therefore, the correct option is \(\boxed{(C)}\).