Classic by-parts workbook. Built-in examples like ∫xⁿeˣ and ∫xⁿln x verify ∫u dv=uv−∫v du numerically, with mnemonics—for by-parts beginners.
Show more ▾
∫ u dv = uv − ∫ v du
Choose a preset to fill in f(x) and interval.
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration
Result
Result ≈
📖 Tutorial | Integration by Parts
1. Definition
When the integrand is a product of two types, use ∫ u dv = uv − ∫ v du, turning the hard part into an easier one. Choose u by LIATE: Log > Inverse-trig > Algebraic > Trig > Exponential.
2. Symbols
Symbols
Meaning
u
Part to differentiate
dv
Part to integrate
v
=∫dv
∫v du
Reduced integral
3. How it works
Choose a classic product example;
Adaptive Simpson directly computes ∫u dv;
Compare with the LIATE analytic result.
4. Steps
Pick a by-parts example;
The integrand and bounds fill in, with u and dv noted;
Click Calculate for the numeric result.
5. Example
Example: ∫₀¹ x eˣ dx. u=x, dv=eˣdx. = [xeˣ]₀¹ − ∫₀¹ eˣ dx = e − (e−1)=1. Tool: 1.000000.
6. Pitfalls
Wrong u/dv makes it worse; follow LIATE; ∫v du still must be integrated; do not drop it; Sometimes parts is needed repeatedly.
❓ FAQ | Integration by Parts
What is LIATE?
Priority for u: Log > Inverse-trig > Algebraic > Trig > Exponential. Choose the earlier as u.
Why integrate ln x by parts?
Write it as 1·ln x with u=ln x, dv=dx; this turns it into a rational integral.
Still not integrable after parts?
Swap u and dv, or try substitution; this tool only verifies numerically.
Does the tool auto-pick u?
No; it offers classic examples and verifies numerically.