Know and be able to use the fundamental theorem of calculus.

i.e. Learners should know that integration may be defined as the reverse of differentiation and be able to apply the result that
\(\displaystyle\int{f(x)}~dx = F(x) + c \iff f(x) = \dfrac{d}{dx}\big(F(x)\big)\),
for sufficiently well-behaved functions.

Includes understanding and being able to use the terms indefinite and definite when applied to integrals.

Fundamental theorem of calculus

Be able to integrate \(x^n\) where \(n \neq 1\) and related sums, differences and constant multiples.

Learners should also be able to solve problems involving the evaluation of a constant of integration e.g. to find the equation of the curve through \((-1, 2)\) for which \(\dfrac{dy}{dx} = 2x+1\).

Integration of standard functions

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

[Integrals of \(\arcsin\), \(\arccos\) and \(\arctan\) will be given if required.]

This includes using trigonometric relations such as double-angle formulae to facilitate the integration of functions such as \(cos^2{x}\).

Integration of standard functions

Be able to evaluate definite integrals.

Definite integration

Be able to use a definite integral to find the area between a curve and the \(x\)-axis.

This area is defined to be that enclosed by a curve, the \(x\)-axis and two ordinates. Areas may be included which are partly below and partly above the \(x\)-axis, or entirely below the \(x\)-axis.

Area under and between curves

Be able to use a definite integral to find the area between two curves.

This may include using integration to find the area of a region bounded by a curve and lines parallel to the coordinate axes, or between two curves or between a line and a curve.

This includes curves defined parametrically.

Area under and between curves

Understand and be able to use integration as the limit of a sum.

In particular, they should know that the area under a graph can be found as the limit of a sum of areas of rectangles.

See also 1.09f.

Integration as the limit of a sum

Be able to carry out simple cases of integration by substitution.

Learners should understand the relationship between this method and the chain rule.

Learners will be expected to integrate examples in the form \(f'(x)\big(f(x)\big)^n\),
such as \((2x + 3)^5\) or \(x(x^2 + 3)^7\), either by inspection or substitution.

Learners will be expected to recognise an integrand of the form \(\dfrac{kf'(x)}{f(x)}\) such as \(\dfrac{x^2+x}{2x^3+3x^2-7}\) or \(\tan{x}\).

Integration by substitution is limited to cases where one substitution will lead to a function which can be integrated. Substitutions may or may not be given.

Learners should be able to find a suitable substitution in integrands such as \(\dfrac{(4x-1)}{(2x+1)^5}\), \(\sqrt{(9-x^2)}\) or \(\dfrac{1}{1+\sqrt{x}}\).

Integration by substitution

Be able to carry out simple cases of integration by parts.

Learners should understand the relationship between this method and the product rule.

Integration by parts may include more than one application of the method e.g. \(x^2\sin{x}\).

Learners will be expected to be able to apply integration by parts to the integral of \(\ln{x}\) and related functions.

[Reduction formulae are excluded.]

Integration by parts

Be able to integrate functions using partial fractions that have linear terms in the denominator.

i.e. Functions with denominators no more complicated than the forms \((ax + b)(cx + d)^2\) or
\((ax + b)(cx + d)(ex + f)\).

Integrate using partial fractions

Be able to 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.

Includes: finding by integration the general solution of a differential equation involving separating variables or direct integration; using a given initial condition to find a particular solution.

Solve differential equations

Be able to interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution.

Includes links to differential equations connected with kinematics.

e.g. If the solution of a differential equation is
\(v = 20 - 20e^{-t}\), where \(v\) is the velocity of a parachutist,
describe the motion of the parachutist.

Solve differential equations