Ftc Part 2

The Fundamental Theorem of Calculus, Part 2

FTC Part 2 provides the practical shortcut for evaluating definite integrals. Instead of computing limits of Riemann sums, you need only find one antiderivative of your integrand and evaluate it at the endpoints:

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

where $F$ is any antiderivative satisfying $F'(x) = f(x)$ on $[a,b]$.

Why the Constant Cancels

Notice that the constant of integration $+C$ disappears: $[F(b)+C] - [F(a)+C] = F(b) - F(a)$

For definite integrals, always use $C=0$ — select the simplest antiderivative.

Evaluation Bar Notation

The standard notation $F(x)\Big|_a^b = F(b) - F(a)$ streamlines writing. It means: evaluate $F$ at the upper limit, then subtract the value at the lower limit.

Worked Examples

Polynomial: Evaluate $\int_1^3 x^2\,dx$.

Since $\frac{d}{dx}\left[\frac{x^3}{3}\right] = x^2$, we have $F(x) = \frac{x^3}{3}$.

$\int_1^3 x^2\,dx = \frac{x^3}{3}\Big|_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$

Trigonometric: Evaluate $\int_0^{\pi/2} \cos x \, dx$.

With $F(x) = \sin x$: $\int_0^{\pi/2} \cos x \, dx = \sin x \Big|_0^{\pi/2} = 1 - 0 = 1$

Common Pitfalls

• Sign errors: Always compute $F(b) - F(a)$, not $F(a) - F(b)$.
• Antiderivative mistakes: Verify by differentiating — confirm $F'(x) = f(x)$.
• Exponent arithmetic: With negative or fractional powers (e.g., $x^{-2}$ has antiderivative $-x^{-1}$), use $\int x^n\,dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$ and recheck your exponent work.

The process is always: find $F(x)$, verify it differentiates to $f(x)$, then compute $F(b) - F(a)$.