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  3. 12th Class
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  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If \[\int_{0}^{x}{f(t)\,dt}=x+\int_{x}^{1}{t\,f(t)\,dt,}\] then the value of \[f(1)\] is [IIT 1998; AMU 2005]

    A) 1/2

    B) 0

    C) 1    

    D) -1/2

    Correct Answer: A

    Solution :

    • \[\int_{0}^{x}{f(t)dt=x+\int_{x}^{1}{t\,f(t)dt\Rightarrow \int_{0}^{x}{f(t)dt=x-\int_{1}^{x}{t\,f(t)dt}}}}\]                   
    • Differentiating w.r.t x, we get \[f(x)=1+\{0-xf(x)\}\]                   
    • Þ \[f(x)=1-xf(x)\Rightarrow (1+x)f(x)=1\Rightarrow f(x)=\frac{1}{1+x}\]                   
    • \ \[f(1)=\frac{1}{1+1}=\frac{1}{2}\].


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