A) \[{{x}^{2}}f(x)+c\]
B) \[|x|\,f(x)+c\]
C) \[|x|\sqrt{f(x)}+c\]
D) \[\frac{2\sqrt{f(x)}}{|x|}+c\]
Correct Answer: D
Solution :
[d] \[I=\int{\frac{xf'(x)-2f(x)}{{{x}^{2}}\sqrt{f(x)}}}\,\,dx=\int{\frac{\frac{{{x}^{2}}f'(x)-2x(x)}{{{x}^{4}}}}{\sqrt{\frac{f(x)}{{{x}^{2}}}}}}dx\] Let \[\frac{f(x)}{{{x}^{2}}}=t.\] \[\Rightarrow \,\,\,I=\int{\frac{dt}{\sqrt{t}}}=2\sqrt{t}+c\] \[=2\sqrt{\frac{f(x)}{{{x}^{2}}}}+c\] \[=\frac{2\sqrt{f(x)}}{|x|}+c\]
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