Integrals

Antiderivatives, the Fundamental Theorem, and the definite integral as area.

An antiderivative of ff is any function FF with F′(x)=f(x)F'(x) = f(x). Antiderivatives differ by a constant, which is why we always write +C+C on indefinite integrals.
∫f(x) dx=F(x)+C\int f(x)\, dx = F(x) + C

The Fundamental Theorem of Calculus

If FF is any antiderivative of ff, then the definite integral of ff from aa to bb equals F(b)−F(a)F(b) - F(a). This is the bridge between antidifferentiation and area under a curve.
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)

The integration techniques you need

  • <strong>Power rule (reversed):</strong> ∫xn dx=xn+1n+1+C\int xn\, dx = \frac{x{n+1}}{n+1} + C for n≠−1n \neq -1.
  • <strong>Constant multiple, sum/difference:</strong> linearity of integration.
  • <strong>uu-substitution:</strong> the chain rule run backwards. Pick u=g(x)u = g(x) so du=g′(x) dxdu = g'(x)\, dx appears in the integrand.
  • <strong>Definite integrals with substitution:</strong> change the bounds to uu values too — don't substitute back.