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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Fundamental Integration

  • question_answer
    \[\int_{{}}^{{}}{{{\{1+2\tan x(\tan x+\sec x)\}}^{1/2}}dx=}\] [Roorkee 1987]

    A)            \[\log (\sec x+\tan x)+c\]         

    B)            \[\log {{(\sec x+\tan x)}^{1/2}}+c\]

    C)            \[\log \sec x(\sec x+\tan x)+c\]

    D)            None of these

    Correct Answer: C

    Solution :

               \[\int_{{}}^{{}}{{{(1+2{{\tan }^{2}}x+2\tan x\sec x)}^{1/2}}dx}\]            \[=\int_{{}}^{{}}{{{({{\sec }^{2}}x+{{\tan }^{2}}x+2\tan x\sec x)}^{1/2}}dx}\] \[=\int_{{}}^{{}}{(\sec x+\tan x)\,dx}=\log (\sec x+\tan x)+\log \sec x+c\] \[=\log \sec x(\sec x+\tan x)+c\].


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