A) \[f(x)=\left( 3{{x}^{2}}-\frac{1}{2x} \right)\]
B) \[f(x)=\frac{1}{5}\,\left( 3{{x}^{2}}-\frac{1}{2x} \right)\]
C) \[a=1\]
D) \[a=2\]
Correct Answer: C
Solution :
Given, \[\int{\frac{12{{x}^{3}}-1}{x\sqrt{100{{x}^{2}}-36{{x}^{6}}-12{{x}^{3}}-1}}dx}\] \[=a\,{{\sin }^{-1}}\,\{f(x)\}+C\] \[\Rightarrow \] \[\int{\frac{6x-\frac{1}{2{{x}^{2}}}}{\sqrt{25-{{\left( 3{{x}^{2}}+\frac{1}{2x} \right)}^{2}}}}dx=a\,{{\sin }^{-1}}\{f(x)\}+C}\] \[=\,\int{\frac{dt}{\sqrt{{{(5)}^{2}}-{{t}^{2}}}}={{\sin }^{-1}}\left( \frac{t}{5} \right)+C}\] \[\left( \text{let}\,\,t=3{{x}^{2}}+\frac{1}{2x} \right)\] \[\Rightarrow \] \[{{\sin }^{-1}}\,\left( \frac{3{{x}^{2}}+\frac{1}{2x}}{5} \right)+C=a\,{{\sin }^{-1}}\,[f(x)]+C\] \[\therefore \] \[a=1,\,\,f(x)=\frac{1}{5}\,\left( 3{{x}^{2}}+\frac{1}{2x} \right)\]
You need to login to perform this action.
You will be redirected in
3 sec
