Need help with this integral

Question

∫ 3x - 5 / x^2 - 25 dx?

thank you!:)

Answer 1

ʃ3x - 5 / x² - 25 dx = ʃ3x - 5 / (x - 5)(x + 5) dx

                            = ʃ(A / x - 5 + B / x + 5) dx

SUB :

3x - 5 / (x - 5)(x + 5) = A(x + 5) + B(x - 5) / (x - 5)( x + 5)

3x - 5 = (A + B)x + 5(A - B)

 A + B  = 3

A - B = -1

Add

2A = 2

A = 1

1 + B = 3

B = 2

Substitute A = 1 and B = 2 in the above integration

                            = ʃ1 / x - 5  dx+ ʃ2 / x + 5 dx

                            = log|x - 5| + 2log|x + 5| + C

                            = log|(x - 5)(x + 5)²| + C