Verified answer

S(g(x) + f(x)) dx = S f(x) dx + S g(x) dx

S 1/(9x² - 1) dx = ?

( S 1/(x² - a²) dx = 1/2a ln|(x - a)/(x +a )| + C

S 1/(9x² - 1) dx = 1/6 ln|(3x - 1)/(3x +1)| + C )

1/(9x² - 1) = a/(3x - 1) + b/(3x + 1) = 1/2(3x - 1) - 1/2(3x + 1)

a(3x + 1) + b(3x - 1) = 1

3ax + a  + 3bx - b = 1

3a + 3b = 0

a - b = 1

a = 1 + b

3 + 3b + 3b = 0

b = -1/2

a = 1/2  

S 1/(9x² - 1) dx = S 1/2(3x - 1) dx - S 1/2(3x + 1) dx = 1/2*1/3 ln(3x - 1) - 1/2*1/3ln(3x + 1) + C = 1/6(ln (3x - 1) - ln(3x + 1)) + C = 1/6 ln|(3x - 1)/(3x +1)| + C

S 1/(a + bx) dx = 1/b ln(a + bx) + C