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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 \[n\] is any integer, then \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\] [IIT 1985; RPET 1995; UPSEAT 2001]

    A) \[x\]                                        

    B) 1

    C) 0    

    D) None of these

    Correct Answer: C

    Solution :

    • Since \[\cos (2n+1)(\pi -x)=\cos [(2n+1)\pi \]\[-(2n+1)x]\]                   
    • = \[-\cos (2n+1)x\] and \[{{\cos }^{2}}(\pi -x)={{\cos }^{2}}x\]           
    • So that \[f(2a-x)=-f(x)\], and hence by the property  of  definite integral \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=0}\].


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