Question

How do you integrate `sec^3x "d"x`?

Answer 1

`intsec^3x"d"x=1/2(secxtanx+lnabs(secx+tanx))+"c"`

Explanation:

For `intsec^3x"d"x`, use integration by parts.

Let `u=secx` and `"d"u=secxtanx"d"x`

and `"d"v=sec^2x"d"x` thus `v=tanx`

Now we plug this into the formula

`intu("d"v)/("d"x)"d"x=uv-intv("d"u)/("d"x)"d"x`

So

`intsec^3x"d"x=secxtanx-intsecxtan^2x"d"x`

`=secxtanx-intsecx(sec^2x-1)"d"x`

`=secxtanx-intsec^3x"d"x+intsecx"d"x`

Thus

`2intsec^3x"d"x=secxtanx+intsecx"d"x`

So

`2intsec^3x"d"x=secxtanx+lnabs(secx+tanx)+"C"`

So

`intsec^3"d"x=1/2(secxtanx+lnabs(secx+tanx))+"c"`

Answer 2

Perform integration by parts, then a substitution.

`int sec^3(x) dx = 1/2(tan(x)sec(x)+lnabs(secx+tanx))+ C`

Explanation:

1. Integrating by parts

Since `sec^3(x) = sec^2(x) sec(x)`,

`int sec^3(x)dx = int sec^2(x) sec(x)dx`

With `u=sec(x) <=> u'=sec(x)tan(x)`, `v'=sec^2(x) <=> v=tan(x)`, and `int uv'dx=uv-int u'vdx`, we have

`int sec^3(x)dx=tan(x)sec(x)-int sec(x)tan^2(x)dx`

Since `sec^2(x)-1=tan^2(x)`,

`int sec(x)tan^2(x)dx`
`= int sec(x)(sec^2(x)-1)dx`
`=int sec^3(x)dx-int sec(x)dx`

Hence

`int sec^3(x)dx`
`=tan(x)sec(x)-(int sec^3(x)dx-int sec(x)dx)`
`=tan(x)sec(x)-int sec^3(x)dx+int sec(x)dx`

If we add `int sec^3(x)dx` to both sides, we have

`2int sec^3(x)dx=tan(x)sec(x)+int sec(x)dx`

`int sec^3(x)dx=1/2(tan(x)sec(x)+int sec(x)dx)`

2. Integrating sec(x)

Now to figure out what `int sec(x)dx` is, you can either look it up in a formula sheet or derive it, as I will now.

Now there are a couple ways to derive this, but I will use the shortest and most common method for this.

`int sec(x)dx=int sec(x)/1dx`

Now if I multiply the numerator and denominator by `sec x+tan x`,

`int sec(x)dx=int (sec x(sec x+tan x))/(sec x + tan x)dx`
`=int (sec^2(x)+secxtanx)/(sec x+ tan x)dx`

Now let `u=sec x+tan x`, thus `du=(secxtanx+sec^2(x))dx`

`int (sec^2(x)+secxtanx)/(sec x+ tan x)dx`

`=int 1/u * (sec^2(x)+secxtanx)dx`

`=int 1/u du = lnabs(u)+C=lnabs(secx+tanx)+C`

3. Formulating the final answer

Hence, given `int sec(x)dx=lnabs(secx+tanx)+C`,

`int sec^3(x)dx=1/2(tan(x)sec(x)+int sec(x)dx)`

`=1/2(tan(x)sec(x)+lnabs(secx+tanx)) + C`