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  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    Let \[f(x)\] be a function satisfying \[{f}'(x)=f(x)\] with \[f(0)=1\] and \[g(x)\] be the function satisfying \[f(x)+g(x)={{x}^{2}}.\] The value of integral \[\int_{0}^{1}{f(x)\,g(x)\,dx}\] is equal to [AIEEE 2003; DCE 2005]

    A) \[\frac{1}{4}(e-7)\]            

    B) \[\frac{1}{4}(e-2)\]

    C) \[\frac{1}{2}(e-3)\]            

    D) None of these

    Correct Answer: D

    Solution :

    • We have \[f'(x)=f(x)\Rightarrow \frac{f'(x)}{f(x)}=1\]                   
    • Þ \[\log f(x)=x+\log c\Rightarrow f(x)=c{{e}^{x}}\]                        
    • Since \[f(0)=1\], therefore \[1=c{{e}^{0}}\Rightarrow c=1\]                   
    • Thus \[f(x)={{e}^{x}}\].                   
    • Hence \[g(x)={{x}^{2}}-{{e}^{x}}\]                   
    • \[\therefore \,\,\int_{0}^{1}{f(x)g(x)dx=\int_{0}^{1}{{{e}^{x}}({{x}^{2}}}-{{e}^{x}})dx}\]                                                                            
    • \[=e-\frac{1}{2}{{e}^{2}}-\frac{3}{2}\].


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