2. Questions Bank
  3. 12th Class
  4. Mathematics
  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If \[2f(x)-3f\left( \frac{1}{x} \right)=x\], then \[\int_{1}^{2}{f(x)}\ dx\] is equal to  [J & K 2005]

    A) \[\frac{3}{5}\ln 2\]             

    B) \[\frac{-3}{5}(1+\ln 2)\]

    C) \[\frac{-3}{5}\ln 2\]           

    D) None of these

    Correct Answer: B

    Solution :

    • \[2f(x)-3f\left( \frac{1}{x} \right)=x\] --(i)           
    • Replacing x by \[\left( \frac{1}{x} \right)\] in (i), we get \[2f\text{ }\left( \frac{1}{x} \right)-3f(x)=\frac{1}{x}\]   --.(ii)           
    • Eliminating \[f\text{ }\left( \frac{1}{x} \right)\] from (i) and (ii), we get           
    • \[-5\ f(x)=2x+\frac{3}{x}=\frac{2{{x}^{2}}+3}{3}\] Þ \[f(x)=-\,\left( \frac{2{{x}^{2}}+3}{5x} \right)\]           
    • \[\int_{1}^{2}{f(x)dx=}\]\[-\int_{1}^{2}{\left( \frac{2{{x}^{2}}+3}{5x} \right)}\ dx=-\frac{1}{5}[{{x}^{2}}+3{{\log }_{e}}x]_{1}^{2}\]                        
    • \[=-\frac{3}{5}[1+{{\log }_{e}}2]=-\frac{3}{5}\left[ 1+\ln 2 \right]\]. 


You need to login to perform this action.
You will be redirected in 3 sec spinner