Question

The value of \(\displaystyle\int_{2}^{2\sqrt3}\frac{dx}{4+x^{2}}\) is:

• A. \(\dfrac{\pi}{12}\)
• B. \(\dfrac{\pi}{18}\)
• C. \(\dfrac{\pi}{24}\)
• D. \(\dfrac{\pi}{6}\)

Hint:

Use \(\int\frac{dx}{a^2+x^2}=\frac1a\tan^{-1}(x/a)\) with \(a=2\).

Answer 1

The Correct Option is C

Step 1: Key Formula:
\(\displaystyle\int\frac{dx}{4+x^{2}}=\frac{1}{2}\tan^{-1}\!\Big(\frac{x}{2}\Big)+C\).

Step 2: Applying the limits:
\(\displaystyle\Big[\frac{1}{2}\tan^{-1}\frac{x}{2}\Big]_{2}^{2\sqrt3}=\frac{1}{2}\Big(\tan^{-1}\sqrt3-\tan^{-1}1\Big)\).

Step 3: Evaluating the inverse tangents:
\(\tan^{-1}\sqrt3=\dfrac{\pi}{3}\) and \(\tan^{-1}1=\dfrac{\pi}{4}\), so the bracket is \(\dfrac{\pi}{3}-\dfrac{\pi}{4}=\dfrac{\pi}{12}\).

Final Answer:
\(\dfrac{1}{2}\times\dfrac{\pi}{12}=\dfrac{\pi}{24}\).\[ \boxed{\dfrac{\pi}{24}} \]