int x^2/(3-x^2) color(red)(dx)∫x23−x2dx
= int ((- 3 + x^2) + 3)/(3-x^2) dx=∫(−3+x2)+33−x2dx
= int -1 + (3)/(3-x^2) dx=∫−1+33−x2dx
= -x + color(blue)( int (3)/(3-x^2) dx) star=−x+∫33−x2dx⋆
For the blue bit, we will use the hyperbolic identity:
1 - tanh^2 y = sech^2 y1−tanh2y=sech2y
So we let x = sqrt 3 tanh y implies dx =sqrt 3 sech^2 y dyx=√3tanhy⇒dx=√3sech2ydy so the blue part of star⋆ becomes:
int (3)/(3-3 tanh^2 y) sqrt 3 sech^2 y \ dy
= int (3)/(3-3 tanh^2 y) sqrt 3 sech^2 y \ dy
= int 1/sech^2 y sqrt 3 sech^2 y \ dy
= sqrt 3 int dy
= sqrt 3 tanh^(-1) (x/ (sqrt 3)) + C
Feeding this into star
implies - x + sqrt 3 tanh^(-1) (x/ (sqrt 3)) + C
