A-Level Maths: Integration Workout

Knowing the integration methods is only half the job: A level Mathematics papers expect you to carry them through quickly and accurately to an exact answer or a numerical estimate. This material is a set of worked integration calculations, based on the integration and numerical methods content of the DfE A level Mathematics subject content.

The first group practises integration by recognition, often called the reverse chain rule. You integrate $(ax+b{)}^n$, spot that $x(x^2+3{)}^4$ is a multiple of a chain-rule derivative, and use the standard forms $\int \frac{f^{\mathrm{\prime }}(x)}{f(x)}dx=\ln \mathrm{\mid }f(x)\mathrm{\mid }+c$ and $\int f^{\mathrm{\prime }}(x)e^{f(x)}dx=e^{f(x)}+c$. Standard trigonometric results such as $\int {\sec }^{2}axdx=\frac{1}{a}\tan ax+c$ and $\int \cos axdx=\frac{1}{a}\sin ax+c$ are used in definite integrals with exact values.

The second group integrates rational functions by first splitting them into partial fractions, giving logarithms that you combine with the laws of logarithms into a single exact answer such as $\ln 2$ or $\ln 6$. The third group finds areas: between a curve and a line, and between two curves, by locating the intersection points and integrating the upper function minus the lower one, and it shows why an integral can be zero while the area it represents is not.

The last group uses the trapezium rule on a table of values. You apply $\frac{h}{2}[y_0+y_n+2(y_1+\dots +y_{n-1})]$ carefully, decide from the sign of the second derivative whether the curve is convex or concave and therefore whether the estimate is too large or too small, and compare an estimate with the exact value to find the percentage error.

The material offers three formats. The quiz has 12 worked multiple-choice questions whose explanations check each answer by differentiation or show the exact arithmetic, and explain the typical slips behind the wrong options. The flashcards collect the standard integrals and the rules for areas and for the trapezium rule. The written work gives 8 longer calculations to set out by hand, including $\int \tan xdx$, a definite integral through partial fractions and a trapezium-rule estimate with its percentage error.

• Integrate by recognition using the reverse chain rule
• Use the standard integrals of sec²ax, cos ax, sin ax, f'(x)/f(x) and f'(x)e^f(x)
• Evaluate definite integrals exactly
• Integrate rational functions using partial fractions
• Find the area between a curve and a line or between two curves
• Apply the trapezium rule and decide whether it overestimates or underestimates

Practice material written by Zestly, based on the DfE A level Mathematics subject content (integration: standard functions, integration by inspection, integration using partial fractions, areas between curves; numerical methods: the trapezium rule).