`int xarcsec(x^2+1) dx` Use integration tables to find the indefinite integral.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Indefinite integral are written in the form of `int f(x) dx = F(x) +C`

 where: `f(x)` as the integrand

           `F(x)` as the anti-derivative function 

           `C`  as the arbitrary constant known as constant of integration

For the given problem `int xarcsec(x^2+1) dx,` it has a integrand in a form of  inverse secant function. The integral resembles one of the formulas from the integration as :  `int arcsec (u/a)du = u*arcsin(u/a) +-aln(u+sqrt(u^2-a^2))+C` .

where we use: `(+)`  if `0ltarcsec (u/a)ltpi/2`

                    `(-)` if `pi/2ltarcsec(u/a)ltpi`

Selecting the sign between `(+)` and` (-) ` will be crucial when solving for definite integral with given boundary values `[a,b]` .

 For easier comparison, we may apply u-substitution by letting:

`u =x^2+1` then `du = 2x dx ` or `(du)/2`

Plug-in the values `int xarcsec(x^2+1) dx` , we get:

`int xarcsec(x^2+1) dx=int arcsec(x^2+1) * xdx`

                                        `= int arcsec(u) * (du)/2`

Apply the basic properties of integration: `int c*f(x) dx= c int f(x) dx` .

`int arcsec(u) * (du)/2= 1/2int arcsec(u) du`

                         or `1/2 int arcsec(u/1) du`

Applying the aforementioned formula from the integration table, we get:

`1/2 int arcsec(u/1) du=1/2 *[u*arcsin(u/1) +-1ln(u+sqrt(u^2-1^2))]+C`

                                `=1/2 *[u*arcsin(u) +-ln(u+sqrt(u^2-1))]+C`

                                `=(u*arcsin(u))/2 +-(ln(u+sqrt(u^2-1)))/2+C`

Plug-in `u =x^2+1` on `(u*arcsin(u))/2 +-(ln(u+sqrt(u^2-1)))/2+C` , we get the indefinite integral as:

`int xarcsec(x^2+1) dx=((x^2+1)*arcsin(x^2+1))/2 +-(ln(x^2+1+sqrt((x^2+1)^2-1)))/2+C`

`=(x^2arcsin(x^2+1))/2+arcsin(x^2+1)/2 +-ln(x^2+1+sqrt(x^4+2x^2))/2+C`

`=(x^2arcsin(x^2+1))/2+arcsin(x^2+1)/2 +-ln(x^2+1+sqrt(x^2(x^2+2)))/2+C`

`=(x^2arcsin(x^2+1))/2+arcsin(x^2+1)/2 +-ln(x^2+1+|x|sqrt(x^2+2))/2+C`



See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial