Integration — OCR A-Level Mathematics

Integration is a fundamental concept in A-Level Mathematics, serving as the inverse operation to differentiation. It allows you to calculate the area under a curve, the total change from a rate of change, and volumes of revolution. In OCR A-Level Mathematics, integration is introduced in Pure Mathematics and extended in Year 2 to include techniques such as integration by substitution, integration by parts, and the integration of standard functions like exponentials and trigonometric functions.

Mastering integration is crucial for success in both Pure and Applied Mathematics. It appears in mechanics when finding displacement from velocity, and in statistics when working with probability density functions. Beyond exams, integration is essential for fields like physics, engineering, and economics. Understanding integration deepens your grasp of calculus and prepares you for further study in mathematics or related disciplines.

In the OCR specification, you will first learn to integrate polynomials and simple functions, then progress to more complex methods. You will also learn to evaluate definite integrals (with limits) and indefinite integrals (with a constant of integration). The topic culminates in applications such as finding areas between curves and volumes of revolution, which are common in exam questions.

• →Indefinite integration: the reverse of differentiation, expressed as ∫f(x) dx = F(x) + c, where F'(x) = f(x) and c is the constant of integration.
• →Definite integration: evaluating ∫ₐᵇ f(x) dx to find the exact area under the curve between x=a and x=b, using F(b) - F(a).
• →Integration of standard functions: including polynomials (∫xⁿ dx = xⁿ⁺¹/(n+1) + c, n≠-1), exponentials (∫eˣ dx = eˣ + c), and trigonometric functions (∫sin x dx = -cos x + c, ∫cos x dx = sin x + c).
• →Integration by substitution: a technique to simplify integrals by changing the variable, often used for composite functions.
• →Integration by parts: based on the product rule, ∫u dv = uv - ∫v du, used for products of functions.

• Correct use of the constant of integration in indefinite integrals.
• Correct application of the fundamental theorem of calculus for definite integrals.
• Correct use of trigonometric identities to facilitate integration.
• Correct application of integration by parts, including multiple applications where necessary.
• Correct identification and use of suitable substitutions for integration.
• Correct separation of variables in differential equations and inclusion of the constant of integration.
• Correct use of partial fractions to integrate rational functions.
• Correct evaluation of areas between curves, including identifying intersection points.

• 💡Always write down the integral expression before using a calculator to evaluate it.
• 💡Check if a trigonometric identity can simplify the integrand before attempting complex methods.
• 💡When solving differential equations, ensure the constant of integration is added before rearranging for the dependent variable.
• 💡Sketch the curves when finding the area between them to identify the correct limits and relative positions.
• 💡Use the 'hence' command word as a hint to use the result from the previous part of the question.
• 💡Always simplify your integrand before integrating if possible. For example, expand brackets or rewrite fractions as powers of x to make integration straightforward.
• 💡Check your answer by differentiating it. If you get back the original function, your integration is correct. This is a quick way to catch errors.
• 💡In definite integrals, be careful with signs when substituting limits. Write out F(b) - F(a) clearly and use brackets to avoid mistakes.

• Forgetting the constant of integration in indefinite integrals.
• Incorrectly applying integration by parts (e.g., sign errors or failing to integrate the second part).
• Failing to change limits when using integration by substitution.
• Incorrectly handling the negative sign in the integration of trigonometric functions.
• Errors in algebraic manipulation when using partial fractions.
• Misinterpreting the area between two curves, particularly when curves cross the x-axis or each other.
• Forgetting the constant of integration (+c) in indefinite integrals. Always include it unless the integral is definite.
• Confusing integration with differentiation, e.g., thinking ∫x² dx = 2x + c. Remember: integration increases the power by 1 and divides by the new power.
• Misapplying integration by parts: choosing u and dv incorrectly. Use the LIATE rule (Log, Inverse trig, Algebraic, Trig, Exponential) to prioritise u.