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  • For an arbitrary constant the Integral sec^2 x / (sec x + tan x ) ^ 9/2 equals

For an arbitrary constant the Integral sec^2 x / (sec x + tan x ) ^ 9/2 equals

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Solution:   We have ,

                    I=int fracsec^2x(sec x + an x)^frac92dx

           Put        (sec x + an x)=t        and    (sec x - an x)=frac1t

   Rightarrow               sec x(sec x + an x)dx=dtRightarrow dx=fracdttcdot sec x=frac1tcdot frac12(t+frac1t)dt

  herefore          I=int fracfrac14(t+frac1t)^2t^frac92cdot tcdot frac12(t+frac1t)dt=frac12int t^frac-92dt +frac12int t^frac-132dt+c

  Rightarrow        I=-frac17t^frac-72-frac111t^frac-112+c=-t^frac-112[frac111+frac17t^2]+c

 Rightarrow        I=frac-1(sec x+ an x)^frac112[frac111+frac17(sec x+ an x)^2]+c.

Posted by

Deependra Verma

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