Integration can be viewed as the inverse of differentiation. An indefinite integral \(\int f(x) \, dx\) represents the family of all antiderivatives of \(f\), expressed as \(F(x) + C\), where \(F'(x) = f(x)\) and \(C\) is the constant of integration capturing the vertical shifts in this family. A definite integral \(\int_a^b f(x) \, dx\) has a more concrete meaning: it represents the signed area between the graph of \(f\) and the \(x\)-axis from \(x = a\) to \(x = b\), with regions above the \(x\)-axis contributing positive area and those below contributing negative area. Together, these two notions of integration are connected by the Fundamental Theorem of Calculus.
The Power Rule for integration reverses the corresponding differentiation rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for any \(n \neq -1\). When \(n = -1\), the special case \(\int \frac{1}{x} \, dx = \ln |x| + C\) applies, where the absolute value is needed because \(1/x\) is defined for negative inputs even though \(\ln x\) is not. Other essential antiderivatives include \(\int e^x \, dx = e^x + C\) (the exponential function is its own antiderivative, just as it is its own derivative), \(\int \sin x \, dx = -\cos x + C\), \(\int \cos x \, dx = \sin x + C\), and \(\int \sec^2 x \, dx = \tan x + C\).
The First Fundamental Theorem of Calculus states that if \(f\) is continuous on \([a, b]\) and \(F\) is any antiderivative of \(f\), then \[ \int_a^b f(x) \, dx = F(b) - F(a). \] This allows definite integrals to be evaluated without resorting to limit-of-Riemann-sums calculations. The Second Fundamental Theorem goes further: if \(f\) is continuous and \(F(x) = \int_a^x f(t) \, dt\), then \(F'(x) = f(x)\). In other words, differentiating an integral with a variable upper limit recovers the original integrand. So \(F(x) = \int_0^x \cos(t^2) \, dt\) has derivative \(\cos(x^2)\), even though no elementary antiderivative of \(\cos(t^2)\) is known.
When the upper limit itself is a function of \(x\), the Chain Rule combines with the Second Fundamental Theorem to give \(\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)\). This composite result is essential in many advanced applications. The Fundamental Theorem is regarded as one of the crowning achievements of calculus because it unifies differentiation and integration as inverse processes and provides a practical method for computing definite integrals without Riemann sums, while also establishing that every continuous function has an antiderivative.