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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x+a}+\sqrt{x+b}}}=\]            [AISSE 1989]

    A) \[\frac{2}{3(b-a)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}]+c\]        

    B) \[\frac{2}{3(a-b)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}]+c\]

    C) \[\frac{2}{3(a-b)}[{{(x+a)}^{3/2}}+{{(x+b)}^{3/2}}]+c\]       

    D) None of these

    Correct Answer: B

    Solution :

    • \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x+a}+\sqrt{x+b}}=\int_{{}}^{{}}{\frac{\sqrt{x+a}-\sqrt{x+b}}{(x+a)-(x+b)}\,dx}}\]                                
    • \[=\frac{1}{(a-b)}\int_{{}}^{{}}{{{(x+a)}^{1/2}}dx}-\frac{1}{(a-b)}\int_{{}}^{{}}{{{(x+b)}^{1/2}}dx}\]                                
    • \[=\frac{2}{3(a-b)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}]+c.\]


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