A-Level Maths Edexcel 9MA0

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.

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, $\sin^2{x}$, $\tan^2{x}$, $\cos^2{3x}$.

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 $y = 4 - x^2$

Understand and use integration as the limit of a sum.

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

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.

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).

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.

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.