A) \[(3/5)\] In 2
B) \[(-3/5)\] (\[1+\]In 2)
C) \[(-3/5)\] In 2
D) None of these
Correct Answer: B
Solution :
Given, \[2f(x)-3f\left( \frac{1}{x} \right)=x\] ?.(i) Put \[x=\frac{1}{x}\] in Eq. (i) we get \[2f\left( \frac{1}{x} \right)-3f(x)=\frac{1}{x}\] ?..(ii) On solving Eqs. (i) and (ii), we get \[f(x)=\frac{3+2{{x}^{2}}}{5x}\] Now, \[\int_{1}^{2}{f(x)\,dx=-\int_{1}^{2}{\frac{3+2{{x}^{2}}}{5x}}dx}\] \[=-\int_{1}^{2}{\left( \frac{3}{5x}+\frac{2x}{5} \right)}dx\] \[=-\frac{1}{5}\left[ 3\log x+\frac{2{{x}^{2}}}{2} \right]_{1}^{2}\] \[=-\frac{1}{5}(3\log 2+4-3\log 1-1)\] \[=-\frac{1}{5}(3\log 2+4-0-1)\] \[=-\frac{1}{5}(3\log 2+3)\] \[=-\frac{1}{5}(3\log 2+3)\]
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