Calculus of a Single Variable, Chapter 8, 8.1, Section 8.1, Problem 26

int (1/(2x+5)-1/(2x-5))dx
To solve, express it as difference of two integrals.
= int 1/(2x+5)dx - int 1/(2x-5)dx
Then, apply substitution method.
u=2x+5

du=2dx
1/2du=dx

w=2x-5

dw=2dx
1/2dw=dx

Expressing the two integrals in terms of u and w, it becomes
= int 1/u*1/2 du - int 1/w*1/2dw
=1/2int1/u du- 1/2 int1/w dw
To take the integral of these, apply the formula int 1/x dx = ln|x|+C .
= 1/2 ln|u| - 1/2 ln|w|+C
And, substitute back u= 2x+5 and w=2x-5.
=1/2ln|2x+5|-1/2ln|2x-5|+C
Therefore, int (1/(2x+5)-1/(2x-5))dx=1/2ln|2x+5|-1/2ln|2x-5|+C .