Know and use the Fundamental Theorem of Calculus.

Fundamental theorem of calculus

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

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

Integration of standard functions

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

Definite integration Area under and between curves

Understand and use integration as the limit of a sum.

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).

Integration by substitution Integration by parts

Integrate using partial fractions that are linear in the denominator.

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).

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.

Solve differential equations