Know and use the Fundamental Theorem of Calculus.

Integration as the reverse process of differentiation. Students should know that for indefinite integrals a constant of integration is required.

Fundamental theorem of calculus

Integrate \(x^n\) (excluding \(n = -1\)) and related sums, differences and constant multiples.

For example, the ability to integrate expressions such as
\(\dfrac{1}{2}x^2-3x^{-\frac{1}{2}}\) and \(\dfrac{(x+2)^2}{x^{\frac{1}{2}}}\) is expected.

Given \(f'(x)\) and a point on the curve, students should be able to find an equation of the curve in the form \(y = f(x)\).

Integrate \(e^{kx}\), \(\dfrac{1}{x}\), \(\sin{kx}\), \(\cos{kx}\) and related sums, differences and constant multiples.

To include integration of standard functions such as \(\sin{3x}\), \(\sec^2{2x}\), \(\tan{x}\), \(e^{5x}\), \(\dfrac{1}{2x}\).

Students are expected to be able to use trigonometric identities to integrate, for example, \(\sin^2{x}\), \(\tan^2{x}\), \(\cos^2{3x}\).

Integration of standard functions

Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.

Students will be expected to be able to evaluate the area of a region bounded by a curve and given straight lines, or between two curves. This includes curves defined parametrically.

For example, find the finite area bounded by the curve \(y = 6x - x^2\) and the line \(y = 2x\)

Or find the finite area bounded by the curve \(y = x^2 - 5x + 6\) and the curve \(y = 4 - x^2\)

Definite integration Area under and between curves

Understand and use integration as the limit of a sum.

Recognise \(\displaystyle\int_a^b{f(x)}~dx = \lim\limits_{\delta x→0} \displaystyle\sum_{x=a}^b f(x) \delta x \)

Integration as the limit of a sum

Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively.

(Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.)

Students should recognise integrals of the form

\(\displaystyle\int{\dfrac{f'(x)}{f(x)}}~dx \implies \ln{|f(x)|} + c \)

The integral \(\int{\ln{x}}~dx\) is required.

Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.

Integration by substitution Integration by parts

Integrate using partial fractions that are linear in the denominator.

Integration of rational expressions such as those arising from partial fractions,
e.g. \(\dfrac{2}{3x+5}\)

Note that the integration of other rational expressions,
such as \(\dfrac{x}{x^2+5}\) and \(\dfrac{2}{(2x-1)^4}\) is also required (see previous paragraph).

Integrate using partial fractions

Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions.

(Separation of variables may require factorisation involving a common factor.)

Students may be asked to sketch members of the family of solution curves.

Solve differential equations

Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.

The validity of the solution for large values should be considered.

Solve differential equations