Question

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

Answer 1

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

Explanation:

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`