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: Trigonometric substitution:
Put \(x=2\tan\theta\), so \(dx=2\sec^2\theta\,d\theta\) and \(4+x^2=4\sec^2\theta\).

Step 2: Rewriting the integral:
\(\displaystyle\int\frac{2\sec^2\theta\,d\theta}{4\sec^2\theta}=\frac12\int d\theta=\frac{\theta}{2}=\frac12\tan^{-1}\frac{x}{2}\).

Step 3: Changing the limits and evaluating:
At \(x=2\), \(\theta=\pi/4\); at \(x=2\sqrt3\), \(\theta=\pi/3\). Value \(=\frac12(\pi/3-\pi/4)=\pi/24\).

Final Answer:
\[ \boxed{\dfrac{\pi}{24}} \]