Fundamental theorem of calculus

FTC Part 1 & The Bridge

The Fundamental Theorem of Calculus proves that differentiation and integration are inverse operations.

Part 1 (Derivative of an integral): If $F(x) = \int_{a}^{x} f(t)\,dt$, then:

$$F'(x) = f(x)$$

Common pitfall: The two parts point in opposite directions. Do not mix up accumulation derivatives with antiderivative evaluation.

FTC Part 2 & Techniques

Part 2 (Evaluation): If $F$ is any antiderivative of $f$ on $[a,b]$, then:

$$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$

Core Techniques:

• Substitution: Let $u=g(x)$, then $du=g'(x)\,dx$.
• Integration by parts: $\int u\,dv = uv - \int v\,du$.
• Partial fractions: Decompose rational integrands first.

Improper integrals: Used when limits are $\pm\infty$ or the integrand is unbounded.

$$\int_{1}^{\infty}\frac{1}{x^{2}}\,dx = \lim_{R\to\infty}\left[-\frac{1}{x}\right]_{1}^{R} = 1$$