Question

If \(f(x)\) is an antiderivative of \(\frac{x+1}{\sqrt{x-1}}\) with respect to \(x\), then value of \(f(5)-f(2)\) is...

• A. \(4\)
• B. \(\frac{14}{3}\)
• C. \(7\)
• D. \(\frac{26}{3}\)

Hint:

Substitute u = x - 1 so that the root becomes a simple power.

Answer 1

The Correct Option is D

Step 1: Split the fraction
$\dfrac{x+1}{\sqrt{x-1}} = \dfrac{(x-1) + 2}{\sqrt{x-1}} = \sqrt{x-1} + \dfrac{2}{\sqrt{x-1}}$.

Step 2: Antiderivative
$f(x) = \frac{2}{3}(x-1)^{3/2} + 4(x-1)^{1/2} + C$.

Step 3: Values
$f(5) = \frac{2}{3}\cdot8 + 4\cdot2 + C = \frac{40}{3} + C$. $f(2) = \frac{2}{3} + 4 + C = \frac{14}{3} + C$.

Step 4: Subtract
$f(5) - f(2) = \frac{26}{3}$. The constant $C$ cancels, so the answer does not depend on it.

Final Answer:
The difference is 26/3. This is option (D). \[ \boxed{\text{(D) }\frac{26}{3}} \]