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12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    \[\int_{0}^{1}{\frac{{{x}^{7}}}{\sqrt{1-{{x}^{4}}}}dx}\] is equal to  [AMU 2000]

    A) 1    

    B) \[\frac{1}{3}\]

    C) \[\frac{2}{3}\]                      

    D) \[\frac{\pi }{3}\]

    Correct Answer: B

    Solution :

    • \[I=\int_{0}^{1}{\frac{{{x}^{7}}}{\sqrt{1-{{x}^{4}}}}dx=\int_{0}^{1}{\frac{{{x}^{6}}x\,dx}{\sqrt{1-{{x}^{4}}}}}}\]           
    • Put  \[{{x}^{2}}=\sin \theta \] \[\Rightarrow 2x\,dx=\cos \theta \,d\theta \]           
    • \[I=\frac{1}{2}\int_{0}^{\pi /2}{\frac{{{\sin }^{3}}\theta .\cos \theta \,\,d\theta }{\cos \theta }}=\frac{1}{2}\int_{0}^{\pi /2}{{{\sin }^{3}}\theta \,\,d\theta }\]    
    • \[=\frac{1}{2}\frac{\Gamma 2\,\Gamma (1/2)}{2.\Gamma (5/2)}=\frac{\Gamma \left( \frac{1}{2} \right)}{4.\frac{3}{2}.\frac{1}{2}.\Gamma \left( \frac{1}{2} \right)}=\frac{1}{3}\].


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