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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Fundamental Integration

  • question_answer
    If \[f'(x)=\frac{1}{x}+x\] and \[f(1)=\frac{5}{2}\], then \[f(x)=\]

    A)            \[\log x+\frac{{{x}^{2}}}{2}+2\]

    B)            \[\log x+\frac{{{x}^{2}}}{2}+1\]

    C)            \[\log x-\frac{{{x}^{2}}}{2}+2\]

    D)            \[\log x-\frac{{{x}^{2}}}{2}+1\]

    Correct Answer: A

    Solution :

               \[f(x)=\int_{{}}^{{}}{{f}'(x)\,dx}=\int_{{}}^{{}}{\left( \frac{1}{x}+x \right)}\,dx=\log x+\frac{{{x}^{2}}}{2}+c\]            Put \[x=1,\] then \[\frac{5}{2}=0+\frac{1}{2}+c\Rightarrow c=2\]            Therefore, \[f(x)=\log x+\frac{{{x}^{2}}}{2}+2.\]


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