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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    The value of \[\int{{{\sec }^{3}}x\,\,dx}\] will be  [UPSEAT 1999]

    A) \[\frac{1}{2}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\]

    B) \[\frac{1}{3}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\]

    C) \[\frac{1}{4}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\]

    D) \[\frac{1}{8}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\]

    Correct Answer: A

    Solution :

    • Let \[I=\int{{{\sec }^{3}}xdx}\]\[=\int{\sec x{{\sec }^{2}}xdx}\]           
    • \[\Rightarrow I=\sec x\tan x-\int{\sec x\,{{\tan }^{2}}x\,dx}\]           
    • \[\Rightarrow I=\sec x\tan x-\int{\sec x\,({{\sec }^{2}}x-1)dx}\]           
    • \[\Rightarrow I=\sec x\tan x-\int{{{\sec }^{3}}x\,dx}+\int{\sec x\,dx}\]           
    • \[\Rightarrow \ I=\sec x\tan x-I+\log \,(\sec x\,+\tan x\,)\]           
    • \[\Rightarrow 2I=\sec x\tan x+\log (\sec x+\tan x)\]                
    • \[\Rightarrow I=\frac{1}{2}[\sec x\tan x+\log (\sec x+\tan x)]\].


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