Some integrals with problematic evaluation

[quinyu]

Code:

int((2x-5)/(3x^2-2),x)

This integral is of the

Code:

(d+ex)/(a+cx^2)

general form. IF cd^2+ae^2 is non-zero (in our case, a=-2, c=3, d=-5, e=2, so the expression is 67, non-zero), AND -a*c gives a nice square root (not negative, etc), then let q=sqrt(-a*c), and then split the integral into

Code:

(e/2+cd/2q)*int(1/(cx-q),x)+(e/2-cd/2q)*int(1/(cx+q),x)

In our case, that gives

Code:

(1-15/(2sqrt(6)))int(1/(3x-sqrt(6)),x)+(1+15/(2sqrt(6)))int(1/(3x+sqrt(6)),x)

This can be resolved by the fact that

Code:

int(1/(a+bx),x)=ln(a+bx)/b

So if we plug that all back, we get after simplification

Code:

(4-5sqrt(6))/12*ln(sqrt(2)-x*sqrt(3))+(4+5sqrt(6))/12*ln(sqrt(2)+x*sqrt(3))

∎