What is the integral of $sec^3(x)$ ?

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Answer 1 · 2014-12-22T06:53:23

$I=int sec^3x dx$

by Integration by Pats with:
$u= secx$ and $dv=sec^2x dx$
$=> du=secx tanx dx$ and $v=tanx$ ,

$=secxtanx-int sec x tan^2x dx$

by $tan^2x=sec^2x-1$

$=secxtanx-int (sec^3x-secx) dx$

since $int sec^3xdx=I$ ,

$=secxtanx-I+int sec x dx$

by adding $I$ and $int sec x dx=ln|secx+tanx|+C_1$

$=>2I=secxtanx+ln|secx+tanx|+C_1$

by dividing by 2,

$=>I=1/2secxtanx+1/2ln|secx+tanx|+C_1/2$

Hence,

$int sec^3 dx=1/2secxtanx+1/2ln|secx+tanx|+C$

I hope that this was helpful.

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