Home

Integration by Parts Formula Online Calculator

Integration-by-parts verifier. Enter u(x) and dv(x) to numerically check ∫u dv=uv−∫v du, with steps—for by-parts practice.

Show more ▾
∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
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

📖 Tutorial | Integration by Parts Formula

1. Definition
The by-parts formula: ∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du. It splits a product integral into a boundary term and an easier integral.
2. Symbols
SymbolsMeaning
u(x)Factor differentiated
dv/dxDerivative of v, i.e. dv=v′dx
v(x)Antiderivative of dv
[uv]ₐᵇBoundary uv(b)−uv(a)
∫v duReduced integral
3. How it works
  • Enter u(x) and v′(x)=dv/dx;
  • Numerically take v(x)=V(x)=∫ₐˣ v′(t)dt from a;
  • Compute LHS ∫u·v′ and RHS [uv]−∫v·u′ and check they agree.
4. Steps
  1. Enter the differentiated factor u(x), e.g. x;
  2. Enter v′(x) in dv/dx, e.g. exp(x);
  3. Enter a,b and click Calculate to compare both sides.
5. Example
Example: ∫₀¹ x eˣ dx. u=x, v′=eˣ. LHS=1; RHS=[xeˣ]₀¹−∫₀¹ eˣ dx=1. Both sides agree.
6. Pitfalls
dv/dx is the derivative of v, not v itself;
u′ is a central difference; avoid 0 and singularities;
If both sides differ, check singularities or an overly wide interval.

❓ FAQ | Integration by Parts Formula

Why verify both sides?
Integration by parts is an identity; numeric verification builds intuition for the boundary-minus-integral structure.
What do I enter for dv/dx?
Enter the derivative v′; e.g. for v=eˣ enter exp(x).
How is v(x) obtained?
Numerically V(x)=∫ₐˣ v′(t)dt, an antiderivative anchored at a.
Why are both sides not exactly equal?
u′ is a difference quotient and integration is adaptive Simpson; error is ~1e-9, normal.
Which functions?
+−*/^, sin cos exp log sqrt abs etc.