Fundamental Theorem of Calculus

Overview

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration, showing they are inverse operations.

Part 1: Differentiation of Integrals

If $f$ is continuous on $[a, b]$, then the function:

$g(x) = \int_a^x f(t) \, dt$

is continuous on $[a, b]$, differentiable on $(a, b)$, and:

$g'(x) = f(x)$

Or equivalently:

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

With Variable Upper Limit

If the upper limit is a function $u(x)$:

$\frac{d}{dx} \left[\int_a^{u(x)} f(t) \, dt\right] = f(u(x)) \cdot u'(x)$

With Variable Lower Limit

$\frac{d}{dx} \left[\int_{v(x)}^b f(t) \, dt\right] = -f(v(x)) \cdot v'(x)$

Both Limits Variable

$\frac{d}{dx} \left[\int_{v(x)}^{u(x)} f(t) \, dt\right] = f(u(x)) \cdot u'(x) - f(v(x)) \cdot v'(x)$

Part 2: Evaluation of Integrals

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then:

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

This is often written as:

$\int_a^b f(x) \, dx = \left[F(x)\right]_a^b = F(x)\Big|_a^b$

The Connection

• Part 1: Differentiation undoes integration
• Part 2: Integration undoes differentiation (up to a constant)

Together: $\int_a^x f(t) \, dt$ is an antiderivative of $f(x)$

Examples

Example 1: FTC Part 1 (basic)

$\frac{d}{dx} \left[\int_0^x \cos(t) \, dt\right] = \cos(x)$

Example 2: FTC Part 1 (with chain rule)

$\frac{d}{dx} \left[\int_1^{x^2} e^t \, dt\right] = e^{x^2} \cdot 2x = 2xe^{x^2}$

Example 3: FTC Part 1 (both limits variable)

$\frac{d}{dx} \left[\int_x^{x^3} \sin(t) \, dt\right] = \sin(x^3) \cdot 3x^2 - \sin(x) \cdot 1$

$= 3x^2 \sin(x^3) - \sin(x)$

Example 4: FTC Part 2 (polynomial)

$\int_1^3 (2x - 1) \, dx = \left[x^2 - x\right]_1^3$

$= (9 - 3) - (1 - 1) = 6$

Example 5: FTC Part 2 (exponential)

$\int_0^{\ln(2)} e^x \, dx = \left[e^x\right]_0^{\ln(2)}$

$= e^{\ln(2)} - e^0 = 2 - 1 = 1$

Example 6: FTC Part 2 (trigonometric)

$\int_0^{\pi/4} \sec^2(x) \, dx = \left[\tan(x)\right]_0^{\pi/4}$

$= \tan\left(\frac{\pi}{4}\right) - \tan(0) = 1 - 0 = 1$

Example 7: Finding a function

If $\int_0^x f(t) \, dt = x^2 + 2x$, find $f(x)$.

By FTC Part 1:

$f(x) = \frac{d}{dx}[x^2 + 2x] = 2x + 2$

Check: $\int_0^x (2t + 2) \, dt = \left[t^2 + 2t\right]_0^x = x^2 + 2x$ ✓

Example 8: Proof that integration is inverse of differentiation

Let $F(x) = \int_a^x f(t) \, dt$

By FTC Part 1: $F'(x) = f(x)$

Therefore $F$ is an antiderivative of $f$.

By FTC Part 2:

$\int_a^b f(x) \, dx = F(b) - F(a) = \int_a^b f(t) \, dt - \int_a^a f(t) \, dt$

This confirms the two parts are consistent.

Mean Value Theorem for Integrals

If $f$ is continuous on $[a, b]$, there exists $c$ in $[a, b]$ such that:

$\int_a^b f(x) \, dx = f(c)(b - a)$

This connects to the average value: $f(c) = f_{\text{avg}}$