Revise Integration as antidifferentiation for IB Mathematics: Analysis and Approaches (5.5)

If differentiating $F(x)$ gives $f(x)$, then $F(x)$ is an antiderivative of $f(x)$, and finding it is integration. We write this using the indefinite integral.

$\int f(x)\,dx = F(x) + C$

The symbol $\int$ is the integral sign, $f(x)$ is the integrand, and $dx$ tells us we are integrating with respect to $x$. The result $F(x)+C$ is the most general antiderivative, where $F'(x)=f(x)$ and $C$ is the constant of integration. The word 'indefinite' signals that we are not yet pinning down a single function — we are describing a whole family at once.

Just as differentiation has a power rule, so does integration — and it is the differentiation rule run in reverse. To differentiate $x^n$ we multiply by the power and subtract one from it. To integrate we do the opposite, in the opposite order: add one to the power, then divide by the new power.

For any rational $n\neq-1$: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ and with a constant multiplier: $\int ax^n\,dx = \frac{ax^{n+1}}{n+1} + C$

Not everything is a power of $x$. Four standard integrals appear constantly and are simply the reverse of derivatives you already know. Learn them as a set — they are quoted, not derived, in an exam.

$\int e^x\,dx = e^x + C$ $\int \frac{1}{x}\,dx = \ln|x| + C$ $\int \cos x\,dx = \sin x + C$ $\int \sin x\,dx = -\cos x + C$

An indefinite integral gives a family of functions, not one. For instance $x^2+1$, $x^2-10$ and $x^2+50$ all differentiate to $2x$; the constant $C$ is what distinguishes them, shifting the curve vertically. To select the single curve we want, we need one extra piece of information — a point the curve passes through, called a boundary condition or initial condition. Substituting that point lets us solve for $C$ and obtain the particular solution.

In Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and follow-through (FT) means a wrong number early on need not cost you the marks that depend on it, provided your later steps are correct on your own figure. But that protection only exists if your method is written down. Study how each mark below is earned by a specific line: the integration, the constant, the substitution of the boundary condition, and the final equation.

Antidifferentiation is the engine of the rest of calculus. Once you can reverse the power rule and quote the standard integrals, definite integrals attach limits to find exact areas under curves, and boundary conditions reappear whenever a real quantity is reconstructed from its rate of change — displacement from velocity, or volume from a flow rate. Master the habit here — add one and divide, quote the standard integrals, never forget $+C$, and use the given point to pin it down — and the integration that follows becomes variations on a method you already own.