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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration by Substitution

  • question_answer
    \[\int_{{}}^{{}}{\sec x\log (\sec x+\tan x)\ dx=}\]

    A)            \[{{[\log (\sec x+\tan x)]}^{2}}+c\]  

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

    C)            \[{{\sec }^{2}}x+\tan x\sec x+c\]     

    D)            None of these

    Correct Answer: B

    Solution :

                       Let \[\log (\sec x+\tan x)=t\Rightarrow \sec x\,dx=dt\]            Therefore \[\int_{{}}^{{}}{\sec x\,\log (\sec x+\tan x)\,dx}=\int_{{}}^{{}}{t\,dt}\]                                                               \[=\frac{{{t}^{2}}}{2}+c=\frac{{{[\log (\sec x+\tan x)]}^{2}}}{2}+c.\]


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