Question

\( \int \frac{1}{\sqrt{9+8x-x^2}}\,dx = \varphi(x) + c \), then \( \varphi(x) = \)

• A. \( \frac{1}{5}\sin^{-1}\left(\frac{x-4}{5}\right) \)
• B. \( \sin^{-1}\left(\frac{x-4}{5}\right) \)
• C. \( \frac{1}{10}\log\left|\frac{4-x}{4+x}\right| \)
• D. \( \log\left|\frac{4-x}{4+x}\right| \)

Hint:

For integrals involving \( \sqrt{a^2 - (x-h)^2} \), always complete the square and use inverse trigonometric substitution.

Answer 1

The Correct Option is B