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1. LIATE
The LIATE method was first mentioned by Herbert E. Kasube in [1]. The function that appears first in the following list should be u when using integration by parts:
L Logatithmic functions ln(x), log 2 (x), etc. I Inverse trig. functions tan−^1 (x), sin−^1 (x), etc. A Algebraic functions x, 3x^2 , 5x^25 , etc. T Trig. functions cos(x), tan(x), etc. E Exponential functions ex, 2x, etc.
Example 1. ∫ x sin(x)dx.
Following the LIATE method, u = x and dv = sin(x)dx since x is an algebraic function and sin(x) is a trigonometric function. Therefore,
u = x dv = sin(x)dx du = dx v = − cos(x)
and ∫ x sin(x)dx = −x cos(x) −
(− cos(x))dx
= −x cos(x) + sin(x) + C.
WARNING: This technique is not perfect!
There are exceptions to LIATE. Some of these can be solved using the order “ILATE” instead. Sometimes, something completely different needs to be con- sidered.
Example 2. ∫ x^3 ex
2 dx.
Following the LIATE rule, u = x^3 and dv = ex
2 dx. However, we would actually set u = x^2 and dv = xex
2 . u = x^2 dv = xex
2 dx du = 2xdx v = 12 ex 2 1
and so ∫ x^3 ex
2 dx = x^2
ex
2
ex
2 2 xdx
x^2 ex
2 −
xex
2 dx
x^2 ex
2 −
ex
2
=
ex
2 (x^2 − 1) + C.
Example 3. (^) ∫
sec^3 (x)dx.
Following the LIATE rule, u = 1 and dv = sec^3 (x)dx. However, we would actually set u = sec(x) and dv = sec^2 (x).
u = sec(x) dv = sec^2 (x)dx du = sec(x) tan(x)dx v = tan(x) and so ∫ sec^3 (x)dx = sec(x) tan(x) −
tan^2 (x) sec(x)dx
= sec(x) tan(x) −
(sec^2 (x) − 1) sec(x)dx
= sec(x) tan(x) −
sec^3 (x) − sec(x)
dx
= sec(x) tan(x) −
sec^3 (x)dx +
sec(x)dx
= sec(x) tan(x) −
sec^3 (x)dx + ln | sec(x) + tan(x)|.
Since the integral we are solving reappears, we need to add it to the left side to get
2
sec^3 (x)dx = sec(x) tan(x) + ln | sec(x) + tan(x)|.
Finally,
∫ sec^3 (x)dx =
sec(x) tan(x) +
ln | sec(x) + tan(x)| + C.
- Tabular Integration By Parts When integration by parts is needed more than once you are actually doing integration by parts recursively. This leads to an alternative method which just makes the amount of writing significantly less. I will explain this through the following example.
