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`
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