Calculator Input
Use * for multiplication and ^ for powers. Supported functions include sin, cos, tan, sqrt, log, ln, exp, abs, sec, csc, and cot.
Formula Used
For a variable-limit integral, let:
The second fundamental theorem, with the chain rule, gives:
If the lower limit is constant, then u'(x) equals zero. The lower contribution disappears. If the upper limit is x, then v'(x) equals one. The result becomes f(x).
How to Use This Calculator
- Enter the integrand as a function of
t. - Enter the lower limit as a constant or expression in
x. - Enter the upper limit as a constant or expression in
x. - Add the x-value where the derivative should be evaluated.
- Choose radians or degrees for trigonometric inputs.
- Press the calculate button to view theorem steps and results.
- Use the CSV or PDF buttons to save the output.
Example Data Table
| f(t) | Lower u(x) | Upper v(x) | Theorem derivative |
|---|---|---|---|
| t^2 | 0 | x | x^2 |
| sin(t) | 1 | x^2 | sin(x^2) × 2x |
| exp(t) | x | x^2 | exp(x^2) × 2x - exp(x) |
| 1 / (1 + t^2) | 0 | sqrt(x) | [1 / (1 + x)] × [1 / (2sqrt(x))] |
About the Second Fundamental Theorem
Why this theorem matters
The second fundamental theorem links accumulation and instantaneous change. It explains how an integral with a moving boundary changes. This idea appears across calculus, physics, statistics, and engineering. A variable limit acts like a sliding endpoint. As the endpoint moves, the accumulated area also changes.
Single moving limit
The simplest form is easy to remember. If F(x) equals the integral of f(t) from a constant to x, then F'(x) equals f(x). The derivative returns the integrand at the moving endpoint. This works because a small change in x adds a thin strip of area. The strip height is nearly f(x).
Chain rule extension
Many problems use an upper limit like x², sin(x), or sqrt(x). Then the endpoint does not move at unit speed. The chain rule must be included. The derivative becomes f(g(x))g'(x). The calculator handles this by evaluating the integrand at the moving upper limit. It then multiplies by the rate of that limit.
Two moving limits
Advanced problems may move both limits. The upper limit adds area. The lower limit removes area. This creates the formula f(v(x))v'(x) minus f(u(x))u'(x). The minus sign is important. It shows that raising the lower boundary shortens the interval.
Practical calculation
This tool evaluates the theorem numerically. It also estimates the integral itself with Simpson's rule. That extra value helps users compare accumulation and derivative behavior. For best results, write multiplication explicitly. Use radians for most calculus examples unless your problem states degrees. Increase subintervals when the integral changes quickly. Use more decimals when checking homework or technical work.
FAQs
1. What does this calculator find?
It finds the derivative of an integral with variable limits. It also evaluates the limits, endpoint integrand values, theorem contributions, and an approximate integral value.
2. Which theorem form is used?
It uses F'(x) = f(v(x))v'(x) - f(u(x))u'(x). This is the variable-limit form with the chain rule included.
3. Can I use only one moving limit?
Yes. Enter a constant for the other limit. Its derivative becomes zero, so that contribution is removed from the final derivative.
4. What variable should the integrand use?
Use t for the integrand. Use x for the bounds. For example, enter t^2 as the integrand and x^2 as the upper limit.
5. Does the calculator show symbolic simplification?
It shows the theorem structure and numerical evaluation. It does not fully simplify algebraic expressions into a final symbolic form.
6. Why is there a minus sign for the lower limit?
A rising lower limit removes area from the interval. That removal creates the negative lower contribution in the derivative formula.
7. What does Simpson subintervals mean?
It controls the numerical integral estimate. More subintervals can improve accuracy, especially when the integrand bends sharply or changes quickly.
8. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report with inputs, formula, and calculated values.