Step by Step
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The integration by parts formula
∫u dv = uv − ∫v du. This technique is used for integrals that are products of two different types of functions that u-substitution can't handle directly.
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The LIATE priority order
To choose which factor becomes "u," use the LIATE priority order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Whichever category appears first in this list, for the factors in your integral, becomes u; the rest becomes dv.
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Why this specific order works
Logarithmic functions simplify dramatically when differentiated (ln(x) becomes 1/x), so they make excellent choices for u. Exponential functions barely change when differentiated or integrated, making them ideal for dv, which is why they sit at the bottom of the LIATE list.
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You may need to repeat the process
After one application of integration by parts, the resulting integral (∫v du) sometimes still requires another round of integration by parts before it can be fully evaluated.
Applied Walkthrough
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Evaluate ∫x·eˣ dx. Between x (Algebraic) and eˣ (Exponential), Algebraic comes first in LIATE, so let u = x and dv = eˣdx.
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Then du = dx, and v = eˣ (the integral of eˣdx).
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Apply the formula: ∫x·eˣ dx = x·eˣ − ∫eˣ dx.
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The remaining integral is simple: ∫eˣ dx = eˣ + C. So the final answer is x·eˣ − eˣ + C.
Exam Application
Exams test whether you can correctly apply the LIATE priority order to choose u, and whether you can carry the integration by parts formula through to a full solution, including cases requiring a second application.
⚠ Common Trap
The most common trap is choosing u and dv in the wrong order relative to LIATE — picking the wrong u often leads to an integral that's more complicated than the one you started with, rather than simpler.
✓ Quick Self-Check
1. What is the integration by parts formula?
∫u dv = uv − ∫v du.
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2. What does the LIATE mnemonic help you decide?
Which factor in the integral to designate as u, based on priority order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
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3. Evaluate ∫x·eˣ dx.
x·eˣ − eˣ + C.
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4. Why do logarithmic functions make good choices for u?
Because they simplify dramatically when differentiated — for example, ln(x) becomes 1/x.
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5. What should you do if, after one application of integration by parts, the resulting integral is still not solvable directly?
Apply integration by parts again to the remaining integral.
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