A-Level Maths OCR A H240

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.*

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 $\dfrac{dy}{dx} = 2x+1$.

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 $cos^2{x}$.*

Be able to evaluate definite integrals.

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.

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.*

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.*

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 $\dfrac{1}{1+\sqrt{x}}$.*

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.*

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 $(ax + b)(cx + d)(ex + f)$.

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.*

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.*