AP Calculus AB and BC

Integral of ln(x^3): Simplify First

The integral of ln of x cubed is 3 times x ln x minus x, plus C. The logarithm rule rewrites the integrand as 3 ln x, so the exponent becomes a constant multiplier and the problem reduces to the standard integral of ln x.

∫ln⁡(x3) dx=3(xln⁡x−x)+C\int \ln\left(x^{3}\right)\,dx = 3\left(x\ln x - x\right) + C

Pull the exponent out first

ln⁡(x3)=3ln⁡x  ⟹  ∫ln⁡(x3)dx=3∫ln⁡x dx\ln\left(x^{3}\right) = 3\ln x \implies \int \ln\left(x^{3}\right)dx = 3\int \ln x\,dx

The base integral needs parts with u=ln⁡xu = \ln x and dv=dxdv = dx, giving xln⁡x−xx\ln x - x, so the answer is three times that.

Simplifying beats technique

Attacking ln⁡(x3)\ln\left(x^{3}\right) with parts directly works but is slower and invites a chain rule slip in dudu. One line of logarithm algebra removes the difficulty entirely.

Common mistakes

  • Reading ln⁡(x3)\ln\left(x^{3}\right) as (ln⁡x)3\left(\ln x\right)^{3}, which is a completely different integral.
  • Forgetting the −x-x term from the parts result.

Every answer on this page is machine checked

An automated test differentiates the antiderivative above and confirms it returns the integrand. A wrong sign or a missing factor fails the build, so it cannot reach you.

Frequently asked questions

What is the integral of ln(x^3)?

It is 3(xln⁡x−x)+C3\left(x\ln x - x\right) + C.

Can I simplify before integrating?

Yes, and you should: ln⁡(x3)=3ln⁡x\ln\left(x^{3}\right) = 3\ln x turns it into a constant times a standard integral.