Fundamental Theorem Of Calculus

Level 1 - Calculus topic page in Integral Calculus.

Principle

The fundamental theorem connects differentiation and integration. It says accumulation functions have derivatives, and definite integrals can be evaluated using antiderivatives.

Notation

\(A(x)=\int_a^x f(t)\,dt\)

accumulation function

\(t\)

dummy variable of integration

\(F\)

antiderivative of f

The Core Method

There are two linked statements.

FTC Part 1

\[\frac{d}{dx}\int_a^x f(t)\,dt=f(x)\]

FTC Part 2

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

Use Part 1 for derivatives of accumulation functions and Part 2 for evaluating definite integrals.

Worked Cases

Question

Find \(\frac{d}{dx}\int_2^x \cos(t^2)\,dt\).

Answer

The upper limit is \(x\), and the integrand is continuous. By the fundamental theorem, the derivative is the integrand evaluated at \(t=x\): \(\cos(x^2)\).

Examples

Question

Differentiate \(\int_0^{x^2} e^t\,dt\).

Answer

Let \(u=x^2\). The derivative of \(\int_0^u e^t\,dt\) with respect to \(u\) is \(e^u\). Multiply by \(du/dx=2x\). The derivative is \(2xe^{x^2}\).

Definite Integrals

Substitution