Cambridge: Pure Mathematics 2 and 3

This chapter introduces integration, which is the reverse process of differentiation. You will learn to find the 'antiderivative' of a function, a process used to calculate quantities like the area under a graph or the volume of a solid. The topic begins with indefinite integrals, where you find a general function whose derivative is the one you started with, always remembering to add the constant of integration, 'C'. From there, you will move to definite integrals, which involve evaluating the integral between two specific limits to find a numerical value, often representing an area.

Key techniques covered include integrating standard functions like polynomials, exponentials, and trigonometric functions. You will also learn more advanced methods such as integration by substitution, which simplifies complex integrals, and integration by parts, used for products of functions. The chapter applies these skills to practical geometric problems, such as finding the area between a curve and an axis, the area between two curves, and the volume of solids formed by rotating a region around an axis.

The most common difficulty is choosing the correct integration method for a given problem. This requires practice and pattern recognition. A solid understanding of differentiation is fundamental, as you will often need to think backwards. Additionally, strong skills in algebra, particularly with partial fractions, and trigonometry are required for the more complex integrals.

Formulas for Cambridge: Pure Mathematics 2 and 3

Integral of a power of x: $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ For n ≠ -1. C is the constant of integration.
Integral of 1/x: $\int \frac{1}{x} \,dx = \ln|x| + C$ For x ≠ 0.
Integral of the exponential function: $\int e^{ax} \,dx = \frac{1}{a} e^{ax} + C$ For a constant a ≠ 0.
Integral of sine: $\int \sin(ax) \,dx = -\frac{1}{a} \cos(ax) + C$ For a constant a ≠ 0.
Integral of cosine: $\int \cos(ax) \,dx = \frac{1}{a} \sin(ax) + C$ For a constant a ≠ 0.
Integral of secant squared: $\int \sec^2(ax) \,dx = \frac{1}{a} \tan(ax) + C$ For a constant a ≠ 0.
Definite integral: $\int_a^b f(x) \,dx = [F(x)]_a^b = F(b) - F(a)$ Where F'(x) = f(x).
Integration by parts: $\int u \frac{dv}{dx} \,dx = uv - \int v \frac{du}{dx} \,dx$ Used to integrate a product of two functions.
Area under a curve: $A = \int_a^b y \,dx$ Area between the curve y=f(x), the x-axis, and lines x=a, x=b.
Volume of revolution about the x-axis: $V = \pi \int_a^b y^2 \,dx$ Volume formed by rotating y=f(x) around the x-axis.
Volume of revolution about the y-axis: $V = \pi \int_c^d x^2 \,dy$ Volume formed by rotating x=g(y) around the y-axis.