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Free fundamental theorem of calculus calculator. Calculate definite integrals, antiderivatives, and apply FTC with step-by-step solutions. Our calculator uses calculus principles to determine all integral relationships using the fundamental theorem.
Last updated: February 2, 2026
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2.666667
Definite integral value
x³/3
F(x) = x³/3
2.666667
F(b) - F(a)
x³/3 + C
+ C constant
Part 1:
d/dx[∫f(t)dt] = f(x)
Part 2:
∫f(x)dx = F(b) - F(a)
Given: ∫[0 to 2] x^2 dx
Step 1: Find antiderivative F(x) = x³/3
Step 2: Apply FTC Part 2: F(2) - F(0)
Step 3: F(2) = 2³/3 = 2.666667
Step 4: F(0) = 0³/3 = 0.000000
Step 5: Result = 2.666667 - 0.000000 = 2.666667
FTC Part 1
Fundamental Theorem Part 1: d/dx[∫f(t)dt] = f(x)
FTC Part 2
Fundamental Theorem Part 2: ∫f(x)dx = F(b) - F(a)
Antiderivative
Antiderivative calculation using power rule
Formula used
∫f(x)dx = F(b) - F(a)
FTC Part 2 application
Formula used
∫f(x)dx = F(x) + C
Antiderivative calculation
Formula used
d/dx[∫f(t)dt] = f(x)
FTC Part 1 application
Formula used
∫ₐᵇ f(x)dx = F(b) - F(a)
FTC Part 2 application
Formula used
∫xⁿdx = xⁿ⁺¹/(n+1) + C
Power rule integration
Features
Complete Analysis
Comprehensive FTC calculations
For ∫[0 to 2] x² dx using FTC:
Definite Integral
2.667
Antiderivative
x³/3
Our fundamental theorem calculator uses the two parts of the Fundamental Theorem of Calculus to calculate definite integrals, antiderivatives, and apply FTC principles. The calculation applies calculus theory and mathematical relationships to determine all integral properties.
Part 1: d/dx[∫f(t)dt] = f(x)Part 2: ∫f(x)dx = F(b) - F(a)Power Rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + CThese formulas form the foundation of integral calculus and allow determination of all integral relationships using the fundamental theorem. They connect differentiation and integration as inverse operations.
Shows the relationship between derivatives and integrals
The Fundamental Theorem of Calculus is the cornerstone of integral calculus that connects differentiation and integration. It provides the theoretical foundation for calculating areas, solving differential equations, and analyzing accumulation functions. Understanding FTC is essential for advanced calculus and mathematical analysis.
Need help with other calculus calculations? Check out our derivative calculator and area between curves calculator.
Get Custom Calculator for Your PlatformResult: ∫[0 to 2] x² dx = 2.667
The definite integral equals the antiderivative evaluated at the bounds.
d/dx[∫₀ˣ t² dt]
= x²
∫x² dx
= x³/3 + C
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