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  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If \[\int_{0}^{1}{{{e}^{{{x}^{2}}}}(x-\alpha )\,dx=0,}\] then [MNR 1994; Pb. CET 2001; UPSEAT 2000]

    A) \[1<\alpha <2\]                   

    B) \[\alpha <0\]

    C) \[0<\alpha <1\]                   

    D) None of these

    Correct Answer: C

    Solution :

    • \[\int_{0}^{1}{{{e}^{{{x}^{2}}}}}(x-\alpha )\,dx=0\]Þ  \[\frac{1}{2}\int_{0}^{1}{2x.{{e}^{{{x}^{2}}}}dx=\alpha \int_{0}^{1}{{{e}^{{{x}^{2}}}}dx}}\]                   
    • Þ  \[\frac{1}{2}|{{e}^{{{x}^{2}}}}|_{0}^{1}=\alpha \int_{0}^{1}{{{e}^{{{x}^{2}}}}dx}\]Þ \[\frac{1}{2}(e-1)=\alpha \,\int_{0}^{1}{{{e}^{{{x}^{2}}}}dx}\]                   
    • Þ \[\alpha =\frac{\frac{1}{2}(e-1)}{\int_{0}^{1}{{{e}^{{{x}^{2}}}}dx}}>0\] and \[\alpha <1\]. So,  \[0<\alpha <1\].


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