2.4 Integration

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Integrate hyperbolic functions and recognise the standard integral forms involving square roots; use trigonometric or hyperbolic substitutions (including completing the square) to evaluate related integrals.
  2. Derive and use reduction formulae to evaluate definite integrals involving powers of functions.
  3. Understand how the area under a curve can be approximated using rectangles; use this idea to estimate areas, set upper and lower bounds, and derive inequalities or limits of sums.
  4. Use integration to find arc lengths (in Cartesian, parametric, and polar form) and surface areas of revolution (in Cartesian and parametric form).

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Objective 1: Integrating Hyperbolic Functions and Square-Root Forms

What are hyperbolic functions?

Hyperbolic functions look very similar to trigonometric functions (like sin and cos), but they are defined differently. They come up naturally in many integration problems.

The main ones you need are:

FunctionNotationIntegral
Hyperbolic sinesinh xcosh x + c
Hyperbolic cosinecosh xsinh x + c
Hyperbolic tangenttanh xln(cosh x) + c
1/cosh²x = sech²xsech²xtanh x + c
sinh x / cosh²x—−1/cosh x + c (i.e. −sech x + c)

Remember: The derivatives of hyperbolic functions mirror those of trig functions, except there are no minus signs to worry about with sinh and cosh (unlike sin and cos where d/dx(cos x) = −sin x, here d/dx(cosh x) = sinh x — no minus sign).

The Three Key Square-Root Integral Forms

These are standard results you must recognise and memorise:

∫1a2−x2 dx=arcsin⁡ ⁣(xa)+c\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\!\left(\frac{x}{a}\right) + c

∫1x2+a2 dx=sinh⁡−1 ⁣(xa)+c(equivalently, ln⁡ ⁣(x+x2+a2)+c)\int \frac{1}{\sqrt{x^2 + a^2}}\,dx = \sinh^{-1}\!\left(\frac{x}{a}\right) + c \quad \text{(equivalently, } \ln\!\left(x + \sqrt{x^2+a^2}\right) + c\text{)} ∫1x2−a2 dx=cosh⁡−1 ⁣(xa)+c(equivalently, ln⁡ ⁣(x+x2−a2)+c)\int \frac{1}{\sqrt{x^2 - a^2}}\,dx = \cosh^{-1}\!\left(\frac{x}{a}\right) + c \quad \text{(equivalently, } \ln\!\left(x + \sqrt{x^2-a^2}\right) + c\text{)}

Here, a is a positive constant, and c is the constant of integration (the unknown value added whenever you find an indefinite integral).

How to remember which is which:

  • If the square root has the form a² − x² (a number minus x²) → use arcsin (a trig inverse function).
  • If the square root has the form x² + a² (x² plus a number) → use sinh⁻¹ (inverse hyperbolic sine).
  • If the square root has the form x² − a² (x² minus a number) → use cosh⁻¹ (inverse hyperbolic cosine).

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