What is the integration of ....?

Question

What is the integration of In x / (x^3)dx?

Answer 1

∫ ( ln(x)/x^3 dx )

∫ ( ln(x) x^(-3) dx )

Use integration by parts.

Let u = ln(x). dv = x^(-3) dx
du = (1/x) dx. v = (-1/2)x^(-2)

uv - ∫ (v du )

(-1/2)x^(-2)ln(x) - ∫ ( (1/x)(-1/2)x^(-2) dx )

Simplify.

(-1/2)(1/x^2)ln(x) - ∫ ( (1/x)(-1/2)(1/x^2) ) dx

Factor out the (-1/2) from the integral; this changes the outside minus into a plus.

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

Multiply the fractions in the integral.

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

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

This is now easily integrable (reverse power rule).

(-1/2)(ln(x)/x^2) + (1/2)(-1/2)(x^(-2)) + C

(-1/2)(ln(x)/x^2) - (1/4)(1/x^2) + C

Answer 2

Integrate by parts
Int= -1/2x^-2*ln x+1/2 int x^-3 dx = -1/2x^-2(1 +1/2 ln x)

Answer 3

intergration of x/x^3 dx
-lnlxl +c
= -1/x +c