Mathfolis

Integration by Parts

Unit 6 · Integration & Accumulation

Integration by Parts (BC Only)

Integration by parts reverses the Product Rule: ∫ u dv = uv − ∫ v du. The LIATE rule guides the choice of u: Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential — pick the earliest type as u. Sometimes two applications are needed, and occasionally the original integral reappears (cyclic case).

Integration by Parts Formula

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du

LIATE Priority for Choosing u

L → I → A → T → E
ln⁡x⏟L,  arctan⁡x⏟I,  xn⏟A,  sin⁡x⏟T,  ex⏟E\underbrace{\ln x}_{L},\; \underbrace{\arctan x}_{I},\; \underbrace{x^n}_{A},\; \underbrace{\sin x}_{T},\; \underbrace{e^x}_{E}
AP Tip: For ∫ eˣ sin x dx or ∫ eˣ cos x dx: apply IBP twice and you'll get the original integral back — solve algebraically for it.
Type 1

Polynomial × Trig or Exponential

Let u = polynomial (A in LIATE). Apply once or twice until the remaining integral is elementary.

Example 1
Evaluate the integral.
∫xcos⁡x dx\int x\cos x\,dx
Example 2
Evaluate the integral.
∫x2ex dx\int x^2 e^x\,dx
Type 2

Logarithm and Inverse Trig

For ln x or arctan x alone, let u = ln x (or arctan x) and dv = dx.

Example 3
Evaluate the integral.
∫ln⁡x dx\int \ln x\,dx
Example 4
Evaluate the integral.
∫x2ln⁡x dx\int x^2 \ln x\,dx
Type 3

Cyclic Integration (eˣ sin/cos type)

Apply IBP twice — the original integral reappears. Collect it on one side and solve algebraically.

Example 5
Evaluate the integral.
∫excos⁡x dx\int e^x \cos x\,dx
← U-SubstitutionPartial Fractions →