What Is Integration by Parts?
Integration by parts is the technique for integrating products — integrals like ∫x·eˣ dx that no basic rule and no substitution will crack, because the two factors have nothing to do with each other's derivatives. It's the integral-world mirror of the product rule: integrating the product rule (uv)' = u'v + uv' and rearranging gives the famous formula
∫ u dv = uv − ∫ v du
The formula doesn't finish the job by itself; it trades your integral for a different one, ∫v du. The art is choosing which factor to call u (it gets differentiated) and which to call dv (it gets integrated) so that the traded integral is genuinely simpler. Choose well and the new integral yields to basic rules; choose badly and it's worse than what you started with.
How to Use This Calculator
Enter a product like x*sin(x) or x^2*cos(x) — or even the deceptive single function ln(x). The calculator selects u and dv using the standard priority (logarithms first, then polynomial factors — the LIATE heuristic), computes du and v, writes out each application of the formula in full, and repeats the process when one round isn't enough (x²·cos(x) needs two). The final answer is verified by differentiation at sample points before being reported, and the graph shows the integrand alongside its antiderivative.
Worked Example
Integrate ∫x·eˣ dx. Choose u = x (a polynomial — differentiating it simplifies it to 1) and dv = eˣ dx. Then du = dx and v = eˣ. Applying the formula:
∫x·eˣ dx = x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C = (x − 1)eˣ + C
Notice the strategy: the polynomial factor lost a degree when differentiated, while eˣ survived integration unchanged. The traded integral ∫eˣ dx was elementary, and one round finished the job. Check by the product rule: d/dx[(x−1)eˣ] = eˣ + (x−1)eˣ = x·eˣ ✓.
Choosing u: the LIATE Heuristic
When deciding which factor to differentiate (u), the classic priority order is L-I-A-T-E: Logarithmic, Inverse trig, Algebraic (polynomials), Trigonometric, Exponential. Pick u as the factor appearing earliest in that list; dv gets the rest.
| Integral | u (differentiate) | dv (integrate) | Why |
|---|---|---|---|
| ∫x·eˣ dx | x | eˣ dx | Algebraic before Exponential |
| ∫x·ln(x) dx | ln(x) | x dx | Logarithmic beats Algebraic |
| ∫x²·sin(x) dx | x² | sin(x) dx | Algebraic before Trig (two rounds) |
| ∫ln(x) dx | ln(x) | dx | The hidden-dv trick |
The logic behind the order: functions early in LIATE simplify when differentiated (ln x → 1/x) but are awkward to integrate, while functions late in the list integrate effortlessly (eˣ → eˣ). Putting each factor on the side that treats it kindly is the whole game — u should get simpler when differentiated, dv should stay manageable when integrated.
Common Mistakes to Avoid
- Choosing u and dv backwards. Try u = eˣ, dv = x dx on the worked example: the traded integral becomes ∫(x²/2)eˣ dx — strictly harder. If the new integral looks worse, swap the roles and restart.
- Dropping the minus sign. The formula subtracts ∫v du; in multi-round problems the alternating signs compound, and one dropped minus wrecks everything downstream. This calculator tracks the sign through each round explicitly.
- Forgetting dv must include dx. dv is a complete differential; v is its antiderivative. Treating dv as just "the other factor" without integrating it is a category error.
- Not recognizing repeat-and-solve situations. For ∫eˣsin(x)dx, two applications reproduce the original integral — the resolution is algebraic: move it across the equals sign and solve for it. (This tool focuses on the terminating polynomial patterns.)
Real-World Applications
Integration by parts is the master key to the transform methods that dominate engineering mathematics. Every entry in a Laplace transform table — the tool electrical engineers use to turn differential equations into algebra — is an integral ∫f(t)e^(−st)dt computed by parts, and the Fourier coefficients behind signal processing, audio compression, and image encoding are ∫f(x)sin(nx)dx integrals of exactly the polynomial-times-trig form this calculator solves. In probability, expected values like E[X] = ∫x·f(x)dx for exponential and gamma distributions are parts integrals, which is why the mean lifetime of radioactive atoms and of light bulbs alike emerges from an x·e^(−λx) integration.
The technique also proves theorems: the reduction formulas in integral tables, the derivation of Taylor's theorem with remainder, and the "integrate by parts to shift the derivative" move that defines weak solutions in the finite element method — the numerical backbone of structural and fluid simulation — are all applications of the same uv − ∫v du exchange.
Frequently Asked Questions
How is ∫ln(x) dx an integration-by-parts problem when there's no product?
Manufacture one: write it as ln(x)·1 dx with u = ln(x) and dv = dx. Then du = dx/x and v = x, giving x·ln(x) − ∫x·(1/x)dx = x·ln(x) − x + C. The invisible-dv trick also handles inverse trig integrals like ∫arctan(x)dx.
How many rounds of parts might a problem need?
One per polynomial degree: xⁿ against sin, cos, or eˣ takes n rounds, the polynomial shedding a degree each time until it's constant. The steps above show every round; the tabular "DI method" taught in many courses is a bookkeeping shortcut for exactly this cascade.
Is LIATE a rule or just a guideline?
A guideline — a very reliable one for textbook integrals, but not a theorem. Occasional integrals reward breaking it, and some (like ∫eˣsin x dx) work with either choice. When LIATE's suggestion leads somewhere worse, the fallback is simple: try the other assignment.
What's the connection to the product rule, exactly?
Integrate (uv)' = u'v + uv' over an interval: uv = ∫u'v + ∫uv', then rearrange to ∫uv' = uv − ∫u'v — the formula, with dv = v'dx notation. Parts is the product rule read backwards, just as u-substitution is the chain rule read backwards.
When should I reach for parts instead of substitution?
Substitution needs the derivative of one part to appear as a factor; parts doesn't. ∫x·e^(x²)dx has the derivative-factor pattern (u-sub territory); ∫x·eˣ dx doesn't — the x is unrelated to eˣ's inner derivative — so parts it is. Checking for the substitution pattern first is the standard triage.