Fundamental Theorem of Calculus

Part I

• Continuity: If $f(x)$ is continuous on $[a, b]$, then the function defined by the integral of $f$ from $a$ to $x$ is also continuous on $[a, b]$.

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

• Differentiation: The derivative of $g(x)$ with respect to $x$ is the original function $f(x)$.

$$g'(x) = \frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$

Variants of Part I

The variants involve differentiating an integral with variable limits of integration:

1. Upper Limit as a Function of $x$:

$$\frac{d}{dx} \int_a^{u(x)} f(t) \, dt = u'(x) f(u(x))$$

2. Lower Limit as a Function of $x$:

$$\frac{d}{dx} \int_{u(x)}^b f(t) \, dt = -u'(x) f(u(x))$$

3. Both Limits as Functions of $x$:

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

Part II

• Anti-Derivative: If $F(x)$ is an anti-derivative of $f(x)$, meaning $F'(x) = f(x)$, then the integral of $f$ from $a$ to $b$ is the difference between the values of $F$ at these points.

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