Section 7.1

Integration by Parts

Substitution handles the Chain Rule. Integration by Parts handles the Product Rule. It lets us trade a difficult integral for an easier one.

1

The Formula

The Integration by Parts Formula

Pick to be something that gets simpler when differentiated.

The ILATE Rule: How to Choose u

Use ILATE to decide which part of your integrand should be . Priority is from top to bottom — choose the first type that appears.

LetterTypeExamples
IInverse Trigonometric
LLogarithmic
AAlgebraic
TTrigonometric
EExponential

Note: Priority is from top to bottom. The first type in your integrand becomes .

Integration by Parts Visualized

Red (Want) = Total Rect (uv) - Blue (Know)

. We trade the Red area (hard) for the Blue area (easy).
2

Worked Example: x sin(x)

Eliminating the x

Evaluate .

1. Choose u and dv (using ILATE)

We have (Algebraic) and (Trigonometric).
ILATE says: A comes before T, so .
Let .

2. Differentiate/Integrate

3. Apply Formula



3

Level Up Examples

Example A: The Logarithm Trick

Evaluate .

1. Choose u and dv (using ILATE)

We only have (Logarithmic).
ILATE says: L is high priority, so .
The only thing left is (the "invisible 1").

2. Steps:
, .
3. Result:

.

Example B: The Infinite Loop

Evaluate .

1. Round 1 (using ILATE)

We have (Trig) and (Exponential).
ILATE says: T comes before E, so , .

.
2. Round 2:
Use parts on .
.
.
3. Solve for I:

.
5

Practice Quiz

Loading...