This is Chapter 7 – Integration for the Year 12 Specialist Mathematics course.

Integration in Specialist Mathematics extends the techniques developed in Mathematical Methods and requires students to make more independent decisions about how an integral should be approached. Rather than relying on one standard rule, students must examine the structure of an expression, recognise a suitable technique and often complete significant algebra before the integration can begin.

We start by extending the standard rules of integration and working with a broader range of functions. Integrals that lead to inverse trigonometric expressions require students to recognise particular algebraic forms and match them with known results.

Integration by substitution is then developed in greater depth. Students must identify an appropriate substitution, transform every part of the integral consistently and return the final answer to the original variable. Integration by parts introduces a method for integrating products of functions and highlights the close relationship between differentiation and integration.

The chapter then moves into geometric applications. Students calculate areas between two functions, carefully identifying which graph lies above the other and where the relevant curves intersect. Solids of revolution extend this idea into three dimensions, using integration to determine the volume created when a region is rotated around an axis.

These problems often combine function analysis, algebra, graphing and calculus in a single question. Accurate notation and well-organised working are especially important, as small errors in limits, substitutions or function order can affect the entire solution.

Advanced integration is central to university-level mathematics, physics, engineering and mathematical modelling, where accumulated quantities, areas and volumes must be calculated from continuously changing systems.

Subchapters:

• 7A: Rules for integration

• 7B: Integrals with inverse trigonometric functions
• 7C: Integration by substitution
• 7D: Integration by parts
• 7E: The area between two functions
• 7F: Solids of revolution

By the end of this chapter, students will be able to select and apply advanced integration techniques and use them to solve both theoretical and geometric problems.