Leibniz Integral Rule Calculator
Differentiate an integral with respect to a parameter, F(t) = ∫ f(x, t) dx from a(t) to b(t), by Leibniz's rule: boundary terms plus the integral of ∂f/∂t, checked against the slope of F.
About the Leibniz Integral Rule Calculator
Leibniz's rule (differentiation under the integral sign): if F(t) = ∫ₐ₍ₜ₎ᵇ⁽ᵗ⁾ f(x, t) dx, then F′(t) = f(b(t), t)·b′(t) − f(a(t), t)·a′(t) + ∫ₐᵇ ∂f/∂t dx. With constant limits only the last term remains.
It turns hard integrals into easier ones: to find ∫₀¹ (xᵗ − 1)/ln x dx, differentiate in t to get ∫₀¹ xᵗ dx = 1/(t + 1), then integrate back to ln(t + 1).
The formula
d/dt ∫ₐ₍ₜ₎ᵇ⁽ᵗ⁾ f(x, t) dx = f(b, t) b′ − f(a, t) a′ + ∫ₐᵇ ∂f/∂t dx
Worked example
F(t) = ∫₀^(t²) sin x dx
- F′(t) = sin(t²) · 2t − 0
- = 2t sin(t²)
Questions people ask
When can I differentiate under the integral sign?
When f and ∂f/∂t are continuous over the region; for improper integrals the integral of ∂f/∂t must also converge uniformly.
Why is the inner integral numerical?
∂f/∂t may have no elementary antiderivative; its value at a given t is still exact to many digits.