Use integration by parts when an integrand is a product whose factors become simpler after one factor is differentiated. Choose and (dv), then apply
Use integration by parts when an integrand is a product whose factors become simpler after one factor is differentiated. Choose and (dv), then apply
This is Mathematics: Analysis and Approaches HL content from AHL 5.16 Integration by substitution, parts and repeated parts.
Integration by parts reverses the product rule:
Integrating both sides and rearranging produces the integration-by-parts formula. The aim is to transform the original integral into one that is easier to evaluate.
Choose as the factor that usually becomes simpler when differentiated. LIATE provides a useful priority guide:
| Priority | Function type |
|---|---|
| L | Logarithmic, such as |
| I | Inverse trigonometric |
| A | Algebraic, such as |
| T | Trigonometric |
| E | Exponential |
Evaluate .
Choose
Differentiate and integrate (dv):
Substitute into the formula:
The remaining integral is simpler because the algebraic factor has disappeared. Differentiate the answer to check:
For , apply integration by parts repeatedly until the algebraic factor disappears. For a cyclic integral, such as , apply the method twice. The original integral then reappears, so collect those terms and solve algebraically for it.
For a definite integral, either apply the limits after finding , or use the bounded form
Keep the same limits throughout. For example, the antiderivative above gives . This also provides a quick exact-value check. This bounded notation is especially useful when endpoint values are simple.
On AA HL Paper 1 or Paper 2, show the choices of , (dv), (du), and before substitution. This makes the method explicit and can earn a method mark even if later arithmetic is incorrect.
A common misconception is that LIATE is compulsory. It is only a guide: the decisive test is whether the new integral is simpler. Include for every indefinite integral, keep signs accurate, and verify the result by differentiation where practical.