Leibniz Integral Rule

Let F(x) be the integral from a to kx of t² + xt. The lower bound a and multiplier k stay constant while x varies. The upper endpoint moves, and the value of the integrand at each fixed t changes too. Leibniz's rule accounts for both effects.

Experiment with the two contributions

Open the interactive experiment and enter these ordinary decimal values. Read the four numerical cards after each edit; the curve and endpoint markers use the same current inputs. The graph can be panned independently of the calculation.

1. Set Lower bound a to 0, Upper-bound multiplier k to 1 and Parameter x to 1. Check Integral F(x) = 0.8333333333 and Total derivative dF/dx = 2.5.
2. Change Lower bound a to 1. The interval collapses and F becomes 0, but the moving-boundary contribution and total derivative remain 2.
3. Set Lower bound a to 0 and Upper-bound multiplier k to 2. The endpoint moves twice as fast; check the contributions 12 and 2, totaling 14.
4. Set Lower bound a to 1 and Upper-bound multiplier k to 0. Check F = −0.8333333333 and derivative −0.5: only the integrand changes.
5. Restore Lower bound a to 0 and Upper-bound multiplier k to 1, then set Parameter x to 0. All four numbers become 0.
6. Set Parameter x to −1. The interval is Reversed, F = −0.8333333333 and the derivative is 2.5. The calculation keeps the bound orientation.
7. Drag the graph, then select Reset view. Parameter x stays −1 and the numerical results remain unchanged.
8. Clear Parameter x to see the error and hidden dependent results. Enter 1 to restore the calculation, or select Reset inputs to restore all defaults.