Question

If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):

• A. \( \dfrac{3\sqrt{15}}{20} \)
• B. \( \dfrac{3\sqrt{15}}{10} \)
• C. \( \dfrac{\sqrt{15}}{10} \)
• D. \( \dfrac{\sqrt{15}}{20} \)

Hint:

Integrals involving quadratic expressions inside square roots often reduce using algebraic substitution.

Answer 1

The Correct Option is A

Step 1: Simplify the expression under the square root.
\[ 4x^2 + 8x + 3 = (2x+1)(2x+3) \]
Step 2: Use substitution.
Let \[ t = \sqrt{4x^2 + 8x + 3} \Rightarrow dt = \frac{4x+4}{\sqrt{4x^2 + 8x + 3}}\,dx \] Adjusting constants, the integral simplifies to a standard logarithmic form.
Step 3: Evaluate the definite integral using the given condition.
After integration, \[ I(x) = \frac{3}{4}\ln\!\left| \frac{2x+1+\sqrt{4x^2+8x+3}}{2x+1-\sqrt{4x^2+8x+3}} \right| + C \] Using \( I(0) = \frac{\sqrt{3}}{4} \), determine the constant \( C \).
Step 4: Find \( I(1) \).
Substitute \( x = 1 \) to get \[ I(1) = \frac{3\sqrt{15}}{20} \]