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What is the integral of #int sec^3(x)dx#?

Answer 1

Paisley Johnson

#I=1/2[secxtanx+ln|secx+tanx|]+C #

Here,

#I=intsec^3xdx.....to(A)#

#I=intsecx(sec^2x)dx#

Using Integration by parts:

#I=secx color(blue)(intsec^2xdx)-int(secxtanxcolor(blue)( intsec^2xdx))dx#

#I=secx*color(blue)(tanx)-intsecxtanx*color(blue)(tanx)dx#

#I=secxtanx-intsecxtan^2xdx#

#I=secxtanx-intsecx(sec^2x-1)dx#

#I=secxtanx-intsec^3xdx+intsecxdx#

#I=secxtanx-I+color(red)(intsecxdx)....to[use,eqn.(A)]#

#I+I=secxtanx+color(red)(ln|secx+tanx|)+c#

#2I=secxtanx+ln|secx+tanx|+c#

#I=1/2[secxtanx+ln|secx+tanx|]+c/2#

#I=1/2[secxtanx+ln|secx+tanx|]+C ,where, C=c/2#

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Answer 2

Emily Ammerman

The integral of (\int \sec^3(x) , dx) is equal to (\frac{1}{2} (\sec(x) \tan(x) + \ln|\sec(x) + \tan(x)|) + C), where (C) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.