∫ √3 dx / (x(x^2+2))

Decompose 1/(x(x^2+2)) into partial fractions

1/(x(x^2+2)) = A/x + (Bx+C)/(x^2+2)

multiply both sides by x(x^2+2)

1 = A(x^2+2) + (Bx+C)x

Match the coefficient of x^2

0 = A + B

A = -B

Match the coefficient of x

0 = C

Match the coefficient of 1

1 = 2A

A = 1/2

B=-A = -1/2

A=1/2; B=-1/2; C=0

1/(x(x^2+2)) = 1/(2x) - x/(2(x^+2))

√3 ∫ dx/(x(x^2+2)) = (√3/2) ∫ dx/x - (√3/2) ∫ xdx/(x^2+2)

√3 ∫ dx/(x(x^+2)) = (√3/2) ln (x) - (√3/2) ∫ xdx/(x^2+2)

To integrate ∫ x dx/(x^2+2)

Let u=x^2+2

du = 2x dx

x dx = (1/2) du

∫ x dx /(x^2+2) = (1/2) ∫ du/u = (1/2) ln(u) = (1/2) ln(x^2+2)

√3 ∫ dx/(x(x^+2)) = (√3/2) ln (x) - (√3/2)(1/2) ln(x^2+2)

= (√3/2) ln (x) - (√3/4) ln(x^2+2) + C