Question

Evaluate : $\int\frac{1}{\sqrt{9+8x-x^{2}}}dx$

• A. $-sin^{-1}\left(\frac{x-4}{5}\right)+C$
• B. $sin^{-1}\left(\frac{x+4}{5}\right)+C$
• C. $sin^{-1}\left(\frac{x-4}{5}\right)+C$
• D. None of these

Answer 1

Answer: (C)

Let $I =\int\frac{1}{\sqrt{9+8x-x^{2}}}dx$. Then,

$I=\int\frac{1}{\sqrt{-\left\{x^{2}-8x-9\right\}}}dx$

$ \Rightarrow I =\int\frac{1}{\sqrt{-\left\{\left(x-4\right)^{2}-5^{2}\right\}}}dx =\int\frac{1}{\sqrt{5^{2}-\left(x-4\right)^{2}}}dx$

$=sin^{-1}\left(\frac{x-4}{5}\right)+C$