integerate : x tanx?

Answer 1

you cannt use the elementry method to integrate this
see this web how integrate it
http://integrals.wolfram.com/index.jsp

where Li is the polylogarithm
http://mathworld.wolfram.com/Polylogarithm.html

Answer 2

Sometimes you just get a horrendous mess, when you try to find an indefinite integral such as this. The indefinite integral of x Tan(x) is:

- x(1+e^(2ix)) + (i/2)(x²+Polylog(2,-e^(2ix)))

There is no explicit expression for this indefinite integral using "elementary" functions. Also, in spite of the appearance of the imaginary i in this expression, it does drop out of the equation when computed in the definite cases where the result converges.

Answer 3

This is why by parts will not work:
lets first look at
∫tan x dx = ∫ sin x / cos x dx

let w =cos x and dw = -sin x dx, so:
∫tan x dx = -∫dw/w = -ln|w| = -ln|cos x| = ln|sec x|, knowing that we can go back to the original problem:

∫xtan x dx
let u = x then du = dx
let dv = tan x dx then v = ln|sec x|

apply by parts: uv - ∫vdu
= x ln|sec x| - ∫ln|sec x| dx

which is as far as you can get because ∫ln|sec x| does not have a closed form solution, use a numerical method.

Answer 4

use integration by parts

Answer 5

How would you do it by parts????