Fundamentals of Integration

Familiarise yourself with the process of integration, the reverse of differentiation, used to calculate the area under a curve, the length of a curve, the surface area of a solid of revolution and the volume of a solid of revolution.
Understand the concept of a definite integral, which gives a numerical value representing the area under a curve.
Recognise indefinite integrals or antiderivatives, which return a function (or family of functions).

Basic Integrations

Know how to deal with basic integrals such as ∫x^n dx, where n ≠ -1. The formula is ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1, with ‘C’ being the unknown constant of integration.
Understand how to use integral tables and the basic rules of integration to solve advanced problems.

Master the power rule for integration, which states ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.
Be able to utilise the sum rule in integration, which allows the integral of a sum of functions to be taken as the sum of their individual integrals.
Know how to apply the constant multiple rule, where the integral of a constant times a function is the constant times the integral of the function.

Get to grips with more advanced techniques such as integration by substitution, a method used when the standard rules of integration cannot be directly applied.
Understand the approach of integration by parts, a technique inspired by the product rule for differentiation.
Learn how to apply integration by partial fractions, a method used to integrate rational functions (fractions where numerator and denominator are polynomials).

Learn how to apply integration to find areas under curves and between curves.
Understand the use of integration in determining the volumes of solids of revolution.
Discover how to employ integration in the calculation of arc lengths of curves.
Recognise the application of integration in determining centre of mass and moments of inertia.
Appreciate how integration is used in solving differential equations, important tools in describing real-world phenomena.

Regularly practice integration problems which tackle real-world examples and abstract problems alike.
Learn from worked examples and make sure to re-do problems independently to ensure thorough understanding and mastery.
Remain diligent and patient when dealing with challenging integrals; these problems will help sharpen mathematical problem-solving skills and aid progression to higher mathematical concepts.