(x + 7)/(x³ - 4x) ≡ (x + 7)/x(x + 2)(x - 2)

. . . . . . . . . . . .≡ A/x + B/(x + 2 ) + C/(x - 2)

. . . . . . . . . . . .≡ A(x + 2)(x - 2) + Bx(x - 2) + Cx(x + 2)/x(x + 2)(x - 2)

. . . . . . . . . . . .≡ A(x² - 4) + B(x² - 2x) + C(x² + 2x)/(x³ - 4x)

. . . . . . . . . . . .≡ [(A + B + C)x² + 2(C - B)x - 4A]/(x³ - 4x)

Hence:

0x² + 1x + 7 ≡ (A + B + C)x² + 2(C - B)x - 4A

and thus

A + B + C = 0

2(C - B) = 1

-4A = 7

From there

A = -7/4

C = B + 1/2

(-7/4) + B + (B + 1/2) = 0

2B = 7/4 - 1/2 = 5/4

B = 5/8

C = B + 1/2

C = 5/8 + 4/8 = 9/8

Partial fractions of the integrand:

(x + 7)/(x³ - 4x) ≡ -7/4x + 5/8(x + 2 ) + 9/8(x - 2)

Therefore

∫ (x + 7)/(x³ - 4x) = ∫ [-7/4x + 5/8(x + 2 ) + 9/8(x - 2)] dx

. . . . . . . . . . . . = (-7/4) ∫ dx/x + (5/8) ∫ dx/(x + 2) + (9/8) ∫ dx/(x - 2) + C

. . . . . . . . . . . . = (-7/4) ln|x| + (5/8) ln|x + 2| + (9/8) ln|x - 2| + C