A) \[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\]
B) \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\]
C) \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\]
D) \[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\]
Correct Answer: B
Solution :
\[af(x)+bf\left( \frac{1}{x} \right)=\frac{1}{x}-5\] (for each \[x\ne 0\]) ?..(i) Replacing x by \[\frac{1}{x}\]in (i), we get \[af\left( \frac{1}{x} \right)+bf(x)=x-5\] ?..(ii) Eliminating \[f\left( \frac{1}{x} \right)\]from (i) and (ii), we get \[({{a}^{2}}-{{b}^{2}})f(x)=\frac{a}{x}-bx-5a+5b\] Þ \[({{a}^{2}}-{{b}^{2}})\int_{1}^{2}{f(x)}dx=\left[ \left( a\log |x|-\frac{b}{2}{{x}^{2}}-5(a-b)x \right) \right]_{1}^{2}\] \[=a\log 2-2b-10(a-b)-a\log 1+\frac{b}{2}+5(a-b)\] \[=a\log 2-5a+\frac{7}{2}b\] Þ \[\int_{1}^{2}{f(x)dx=\frac{1}{{{a}^{2}}-{{b}^{2}}}\left[ a\log 2-5a+\frac{7}{2}b \right]}\].
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