Question

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

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

Answer 1

Answer: (C)

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

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

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

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