How to solve ∫ sec^3x dx ?

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Answer 1 · 2015-10-01T12:24:52

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

Let's solve $int secxdx$ :

$t=secx+tanx => dt=(secxtanx+sec^2x)dx$

$dt=secx(secx+tanx)dx$

$int secxdx = int secx (secx+tanx)/(secx+tanx) dx=int dt/t$
$int secxdx = ln|secx+tanx| +C$

$I = int sec^3xdx = int sec^2x secxdx$

$u=secx => du=secxtanxdx$
$dv=sec^2xdx => v=int sec^2xdx=tanx$

$int sec^3xdx=secxtanx - int tanx secx tanx dx$

$int sec^3xdx=secxtanx - int tan^2x secx dx$

$int sec^3xdx=secxtanx - int (sec^2x-1) secx dx$

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

$int sec^3xdx=secxtanx - int sec^3x dx + int secx dx$

$int sec^3xdx + int sec^3x dx=secxtanx + ln|secx+tanx|$

$2 int sec^3xdx=secxtanx + ln|secx+tanx|$

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