Question

integral of xtanx

this is not very easy as tending to be

Answer 1

here we can use expansion of tanx
tanx = x + x³/3 + 2x^5/15 + 17x^7/315 + ..∞

now,
I = xtanx
= x { x + x³/3 + 2x^5/15 + 17x^7/315 +.....∞}dx

={ x² + x⁴/3 + 2x^6/15 + 17x^8/315 +......∞}dx

= x².dx + x⁴dx/3 + 2x^6.dx/15 + 17x^7dx/315 + .......∞

= x³/3 + x^5/15+ 2x^7/105 + 17x^8/2520 +..........∞

actually here if we use integration by parts , to get the solution is very tough . so we can use the expansion of trigonometry term and then integrate by using fundamental concepts of integration .