Integration by Parts: The LIATE Rule and Worked Examples
Use ∫u dv = uv − ∫v du and the LIATE rule to choose u, with two fully worked calculus examples.
The formula
- ∫ u dv = uv − ∫ v du
Pick u so that du is simpler than u.
CHOOSE u WITH LIATE
- Logarithmic
- Inverse trig
- Algebraic (polynomial)
- Trigonometric
- Exponential
The first type in this list is usually u.
Example 1
Find ∫ x eˣ dx.
- u = x, dv = eˣ dx, so du = dx and v = eˣ
- ∫ x eˣ dx = x eˣ − ∫ eˣ dx
- = x eˣ − eˣ + C
(x − 1)eˣ + C
Example 2
Find ∫ x cos x dx.
- u = x, dv = cos x dx, so du = dx and v = sin x
- ∫ x cos x dx = x sin x − ∫ sin x dx
- = x sin x + cos x + C
x sin x + cos x + C
Common mistake
Choosing u = eˣ and dv = x dx
That makes the new integral harder.
Check by differentiating your answer.
Some integrals need parts twice.
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