Question

The value of $ \int{\frac{dx}{\sqrt{(3-5x-{{x}^{2}})}}} $ is

• A. $ {{\sin }^{-1}}\left( \frac{2x+5}{\sqrt{37}} \right)+c $
• B. $ {{\sinh }^{-1}}\left( \frac{2x+5}{\sqrt{37}} \right)+c $
• C. $ {{\sin }^{-1}}\left( \frac{2x+5}{37} \right)+c $
• D. None of the above

Answer 1

Answer: (A)

$ \int{\frac{dx}{\sqrt{(3-5x-{{x}^{2}})}}} $
$ =\int{\frac{dx}{\sqrt{-\left[ {{x}^{2}}+5x-3+{{\left( \frac{5}{2} \right)}^{2}}-{{\left( \frac{5}{2} \right)}^{2}} \right]}}} $
$ =\int{\frac{dx}{\sqrt{-\left[ {{\left( x+\frac{5}{2} \right)}^{2}}-\frac{37}{4} \right]}}} $
$ =\int{\frac{dx}{\sqrt{{{\left( \frac{\sqrt{37}}{2} \right)}^{2}}-{{\left( x+\frac{5}{2} \right)}^{2}}}}} $
$ ={{\sin }^{-1}}\left[ \frac{\left( x+\frac{5}{2} \right)}{\left( \frac{\sqrt{37}}{2} \right)} \right]+c $
$ ={{\sin }^{-1}}\left[ \frac{2x+5}{\sqrt{37}} \right]+c $