Integral Using Fundamental Theorem of Calculus Calculator

The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to compute definite integrals using antiderivatives. This calculator helps you apply this powerful theorem to find exact values of integrals.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus, first and second parts, establishes a deep connection between differentiation and integration:

First Part: If a function f is continuous on [a, b] and F is an antiderivative of f on [a, b], then ∫[a to b] f(x) dx = F(b) - F(a).
Second Part: If f is continuous on [a, b], then the function g defined by g(x) = ∫[a to x] f(t) dt has a derivative g'(x) = f(x).

Key Formula

∫[a to b] f(x) dx = F(b) - F(a), where F is the antiderivative of f.

This theorem allows us to evaluate definite integrals by finding antiderivatives, which is often much easier than using Riemann sums or other methods.

How to Use This Calculator

To use the calculator:

Enter the function you want to integrate in the "Function" field (e.g., x², sin(x), e^x).
Specify the lower and upper limits of integration (a and b).
Click "Calculate" to compute the integral using the Fundamental Theorem of Calculus.
Review the result and interpretation.

Note

The calculator assumes the function has an antiderivative. For functions without antiderivatives (like |x|), numerical methods would be needed.

Worked Example

Let's compute ∫[1 to 3] 2x dx using the Fundamental Theorem of Calculus.

Find the antiderivative F(x) of f(x) = 2x. F(x) = x² + C.
Apply the theorem: ∫[1 to 3] 2x dx = F(3) - F(1) = (3²) - (1²) = 9 - 1 = 8.

The exact value of the integral is 8.

Frequently Asked Questions

What if the function doesn't have an antiderivative?: For functions without antiderivatives (like |x|), you would need to use numerical integration methods instead.
Can this calculator handle trigonometric functions?: Yes, the calculator can handle trigonometric functions like sin(x), cos(x), and tan(x).
What if I enter an invalid function?: The calculator will display an error message if the function is invalid or cannot be integrated.
Is the result always exact?: Yes, when an antiderivative exists, the result is exact. For functions without antiderivatives, numerical methods would be needed.