How do you find the antiderivative of $int sec^2xtanx dx$ ?

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Answer 1 · 2016-10-27T16:49:24

$intsec^2xtanxdx=1/2sec^2x+c$

Method 1

now $d/(dx)(secx)=secxtanx$

so $intsec^2xtanxdx=1/2sec^2x+c_1$

since $d/(dx)(sec^2x)=2(secx)xxsecxtanx$

Method 2

$d/(dx)(tan^2x)=2(tanx)xxsec^2x$

by the chain rule

so $intsec^2xtanxdx=1/2tan^2x+c_2$

the 2 results can be shown to be equivalent by use of

$1+tan^2x=sec^2x$

How do you find the antiderivative of int sec^2xtanx dxint sec^2xtanx dx?