How to recognize it

01A product of unlike functions

Polynomial × exponential and polynomial × trig are strong signals.

02A lonely logarithm or inverse trig function

Rewrite ln⁡x\ln x as ln⁡x⋅1\ln x\cdot1, then differentiate the awkward factor.

03Differentiation simplifies one factor

If it makes the new integral worse, rethink the choice.

A simple example

∫xex dx\int xe^x\,dxu=x⟹du=dxu=x\quad\Longrightarrow\quad du=dxdv=ex dx⟹v=exdv=e^x\,dx\quad\Longrightarrow\quad v=e^x
∫xex dx=xex−ex+C\boxed{\int xe^x\,dx=xe^x-e^x+C}

The canonical loop

The integral of sec⁡3x\sec^3x is the memorable case because the original integral returns. That creates an equation you can solve.

∫sec⁡3x dx\int\sec^3x\,dxSee why it brings itself back →

When it is the wrong first move

∫2xcos⁡(x2) dx\int2x\cos(x^2)\,dx

Parts is possible, but it is clumsy. Since 2x2x is the derivative of x2x^2, a direct substitution is the clean first move.

Better choice: u=x2u=x^2

Common mistakes

  • Treating LIATE as a law instead of a useful tie-breaker.
  • Choosing dvdv that has no manageable antiderivative.
  • Forgetting the minus sign in uv−∫v duuv-\int v\,du.
  • Stopping before checking whether the new integral is actually simpler.

Ready to test the instinct?

Choose the method before doing the algebra.

Start practice →