Integrate int_1 ^sqrt3 arctan(1/x)dx?

1 Answer
Sep 25, 2017

int_(1)^(sqrt(3)) \ arctan(1/x) \ dx = (sqrt(3)pi)/6 + 1/2 ln2 - pi/4 ~~ 0.468075 ...

Explanation:

We seek:

I = int_(1)^(sqrt(3)) \ arctan(1/x) \ dx

We can apply Integration By Parts

Let { (u,=arctan(1/x), => (du)/dx,=1/(1+1/x^2)(-1/x^2)=-1/(x^2+1)), ((dv)/dx,=1, => v,=x ) :}

Then plugging into the IBP formula:

int \ (u)((dv)/dx) \ dx = (u)(v) - int \ (v)((du)/dx) \ dx

gives us

int \ (arctan(1/x))(1) \ dx = (arctan(1/x))(x) - int \ (x)(-1/(x^2+1))) \ dx

:. int \ arctan(1/x) \ dx = xarctan(1/x) + int \ x/(x^2+1) \ dx
" " = xarctan(1/x) + 1/2 \ int \ (2x)/(x^2+1) \ dx
" " = xarctan(1/x) + 1/2 ln |x^2+1| + C
" " = xarctan(1/x) + 1/2 ln (x^2+1) + C

And so:

I = [xarctan(1/x) + 1/2 ln (x^2+1) ]_(1)^(sqrt(3))

\ \ = (sqrt(3)arctan(1/sqrt(3)) + 1/2 ln (3+1) ) - (arctan(1) + 1/2 ln (1+1) )

\ \ = (sqrt(3)pi/6 + 1/2 ln4 ) - (pi/4 + 1/2 ln2)

\ \ = sqrt(3)pi/6 + 1/2 2ln2 - pi/4 - 1/2 ln2

\ \ = (sqrt(3)pi)/6 + 1/2 ln2 - pi/4

\ \ ~~ 0.468075 ...