Question

If \(\int \frac{dx}{\sqrt{2ax-x^2}} = f(g(x))+c\), where \(c\) is constant of integration, then \(f(x),g(x)\) are respectively equal to

• A. \(sin^{-1}x,\frac{x-a}{a}\)
• B. \(sin^{-1}x,\frac{x+a}{a}\)
• C. \(tan^{-1}x,\frac{x-a}{a}\)
• D. \(tan^{-1}x,\frac{x+a}{a}\)

Hint:

Complete the square: $2ax-x^2=a^2-(x-a)^2$.

Answer 1

The Correct Option is A

Step 1: Substitute
Let $x-a=a\sin\theta$. Then $dx=a\cos\theta\,d\theta$ and the root becomes $a\cos\theta$.
The integral reduces to $\int d\theta=\theta=\sin^{-1}\frac{x-a}{a}$. Option (A).

Final Answer:
Option (A). \[ \boxed{\text{(A)}} \]