Fundamental theorem of calculus

The FTC bridges differentiation and integration in two parts.

Part 1 (Derivative of an integral)

If $F(x) = \displaystyle\int_{a}^{x} f(t)\,dt$, then

F'(x) = f(x)

Part 2 (Evaluation)

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

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

Integration techniques

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

Improper integrals — When a limit of integration is $\pm\infty$ or the integrand is unbounded:

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

Key insight: The FTC says that computing a definite integral reduces to finding one antiderivative and evaluating it at two points.

Common pitfall: The two parts of the fundamental theorem point in different directions: one says accumulation functions have derivative $f$, the other evaluates integrals from antiderivatives. Keep straight which one a problem invokes.

The Fundamental Theorem of Calculus

The Fundamental Theorem has two parts that link differentiation and integration.

Part I: If $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$.

Part II:

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

where $F$ is any antiderivative of $f$.

This theorem shows that differentiation and integration are inverse operations — the central idea of calculus.