Question

If \[ \int\frac{dx}{\sqrt{4x^{2}+11x+6}} = \frac12\cosh^{-1}\!\left(\frac{f(x)}{5}\right)+C \] and \[ f(1)=19, \] then \(f(2)=\) ?

• A. 11
• B. 15
• C. 23
• D. 27

Hint:

For quadratic expressions under a square root, memorize the standard logarithmic and inverse hyperbolic integral forms. They save a lot of computation.

Answer 1

The Correct Option is D

Concept: For integrals of the form \[ \int\frac{dx} {\sqrt{ax^2+bx+c}}, \] we use the standard result \[ \int\frac{dx} {\sqrt{ax^2+bx+c}} = \frac1{\sqrt a} \cosh^{-1} \left( \frac{2ax+b} {\sqrt{b^2-4ac}} \right)+C. \]

Step 1: Identify coefficients.
Given \[ 4x^2+11x+6. \] Thus \[ a=4,\quad b=11,\quad c=6. \]

Step 2: Compute the discriminant term.
\[ b^2-4ac = 121-96 = 25. \] \[ \sqrt{b^2-4ac}=5. \]

Step 3: Apply the standard formula.
\[ \int \frac{dx} {\sqrt{4x^2+11x+6}} = \frac12 \cosh^{-1} \left( \frac{8x+11}{5} \right)+C. \] Comparing with \[ \frac12 \cosh^{-1} \left( \frac{f(x)}5 \right)+C, \] we obtain \[ f(x)=8x+11. \]

Step 4: Find \(f(2)\).
\[ f(2) = 8(2)+11. \] \[ =16+11. \] \[ =27. \] Hence \[ \boxed{27}. \]