AP Calculus BC: Integration by Parts in 4 Steps (with worked example)

Integration by parts costs students more AP Calc BC points than almost any other technique. Not because the math is hard — the procedure is four mechanical steps — but because students don't have a reliable way to choose which factor to call u and which to call dv. Get that choice wrong and you spiral into worse and worse integrals.

Here's the rule Tae uses with his AP Calc BC students, and a worked example showing exactly how it plays out.

The formula

If you have an integral of a product of two functions, you can rewrite it as:

∫ u dv = uv − ∫ v du

You pick one factor to call u and the other to call dv. Then you differentiate u to get du, integrate dv to get v, and plug everything into the right-hand side. The hope is that ∫ v du is easier than the integral you started with.

The choice: LIATE

The order LIATE tells you which factor to call u. Pick the one that comes earliest in the list:

  • Logarithmic (ln x)
  • Inverse trig (arctan x, arcsin x)
  • Algebraic (x, x², polynomials)
  • Trigonometric (sin x, cos x)
  • Exponential (ex, 2x)

Whatever's left becomes dv. LIATE works because each category gets simpler in a predictable way: logs and inverse trigs simplify when differentiated; exponentials and trig functions cycle when integrated.

Worked example

Evaluate ∫ x · ex dx

Step 1 — Identify the factors: we have x (algebraic) and ex (exponential).

Step 2 — Apply LIATE: Algebraic (A) comes before Exponential (E). So u = x and dv = ex dx.

Step 3 — Compute du and v:

  • du = dx (just differentiate x)
  • v = ex (integrate ex)

Step 4 — Plug in:

∫ x ex dx = uv − ∫ v du = x · ex − ∫ ex dx = x ex − ex + C

Done. The new integral on the right side (∫ ex dx) is trivial, which is exactly what LIATE was supposed to guarantee.

When it doesn't work in one pass

Sometimes you need to apply integration by parts twice — usually when you have two cycling functions like x² sin x or ex cos x. The trick: after the second pass, the original integral reappears on the right side, and you can solve for it algebraically. This is a classic AP Calc BC FRQ trap and worth practicing on its own.

Common mistakes that cost AP exam points

  1. Forgetting the minus sign in front of ∫ v du. The formula is uv minus the integral, not plus.
  2. Re-differentiating instead of integrating dv. When you set dv = ex dx, you need v = ex, not (ex)' = ex. Yes, they're the same in this case — which is why students get sloppy and forget to actually integrate.
  3. Picking u and dv backwards. If you try u = ex and dv = x dx in the example above, you'll get an integral that's harder than the one you started with. LIATE prevents this.
  4. Dropping the +C on indefinite integrals. Easy point on an FRQ.

Practice problems

Try these on your own using LIATE:

  1. ∫ x cos x dx
  2. ∫ ln x dx (hint: u = ln x, dv = dx)
  3. ∫ x² ex dx (two passes)
  4. ∫ ex sin x dx (cycling functions)

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