Consider the following statements, where C is a constant of integration:
(A) \(\int\displaystyle\frac{dx}{x^2+25} = \tan^{-1}\frac{x}{5}+C\)
(B) \(\int\displaystyle\frac{dx}{\sqrt{9-x^2}} = \frac{1}{3}\sin^{-1}\frac{x}{3}+C\)
(C) \(\int\displaystyle\frac{dx}{x^2-4} = \frac{1}{4}\log\left|\frac{x-2}{x+2}\right|+C\)
(D) \(\int\displaystyle\frac{dx}{\sqrt{x^2+4}} = \log\left|x+\sqrt{x^2+4}\right|+C\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C) and (D) only
The special standard forms are
\( \displaystyle \int\frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a}+C \)
\( \displaystyle \int\frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a}+C \)
\( \displaystyle \int\frac{dx}{x^2-a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C \)
\( \displaystyle \int\frac{dx}{\sqrt{x^2+a^2}} = \log\left|x+\sqrt{x^2+a^2}\right|+C \)
So with a = 5, 3, 2, 2 the four integrals are \(\frac{1}{5}\tan^{-1}\frac{x}{5}\), \(\sin^{-1}\frac{x}{3}\), \(\frac{1}{4}\log\left|\frac{x-2}{x+2}\right|\) and \(\log\left|x+\sqrt{x^2+4}\right|\), each plus C.
Checking the statements
(A) \(\int\frac{dx}{x^2+25} = \tan^{-1}\frac{x}{5}+C\) — Incorrect. The factor \(\frac{1}{a} = \frac{1}{5}\) is missing.
(B) \(\int\frac{dx}{\sqrt{9-x^2}} = \frac{1}{3}\sin^{-1}\frac{x}{3}+C\) — Incorrect. The \(\sin^{-1}\) form has no factor \(\frac{1}{a}\); the answer is \(\sin^{-1}\frac{x}{3}+C\).
(C) \(\int\frac{dx}{x^2-4} = \frac{1}{4}\log\left|\frac{x-2}{x+2}\right|+C\) — Correct. With a = 2 the factor is \(\frac{1}{2a} = \frac{1}{4}\).
(D) \(\int\frac{dx}{\sqrt{x^2+4}} = \log\left|x+\sqrt{x^2+4}\right|+C\) — Correct. It is the standard form with a = 2.