\(\int\displaystyle\frac{dx}{\sqrt{4x^2-25}}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{2}\log\left|2x+\sqrt{4x^2-25}\right|+C\)
Let \(t = 2x\), so \(dx = \frac{dt}{2}\) and \(4x^2-25 = t^2-5^2\):
\( \displaystyle \int\frac{dx}{\sqrt{4x^2-25}} = \frac{1}{2}\int\frac{dt}{\sqrt{t^2-5^2}} \)
Using the standard integral \(\int\frac{dt}{\sqrt{t^2-a^2}} = \log\left|t+\sqrt{t^2-a^2}\right|\):
\( \displaystyle = \frac{1}{2}\log\left|2x+\sqrt{4x^2-25}\right|+C \)
Why the other options are incorrect
Option (1) \(\log\left|2x+\sqrt{4x^2-25}\right|+C\) — Incorrect. It drops the factor \(\frac{1}{2}\) from \(dx = \frac{dt}{2}\); its derivative is \(\frac{2}{\sqrt{4x^2-25}}\).
Option (3) \(\displaystyle\frac{1}{2}\log\left|2x-\sqrt{4x^2-25}\right|+C\) — Incorrect. Its derivative is \(-\frac{1}{\sqrt{4x^2-25}}\), the negative of the integrand.
Option (4) \(\displaystyle\frac{1}{10}\log\left|\frac{2x-5}{2x+5}\right|+C\) — Incorrect. This is \(\int\frac{dx}{4x^2-25}\), the integral without the square root.