Integration - Chapter 13 - Integration Cheat Sheet

Oct 13, 2026

Integration is presented as the reverse of differentiation, emphasizing how to compute areas bounded by curves and coordinate axes. The guide explains key notation (the integral symbol, integrand, and limits), distinguishes indefinite from definite integrals, and highlights the role of the constant of integration. It provides worked examples for evaluating common integral forms, handling polynomials, determining the constant from a point on a curve, and finding bounded areas, including cases where curves cross the x-axis.

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Integration Cheat Sheet

Notation

Indefinite integrals

Definite integrals

Integrating polynomials

Finding the constant of integration

Finding areas

Areas under the x-axis

More complicated areas

Integration Cheat Sheet Integration is the reverse of differentiation. We can use integration to find areas bounded between a curve and the coordinate axes. Notation The ∫ symbol is used to represent integration. Since integration is the reverse of differentiation, we have that The 􀀡􀀢 means that we are integrating with respect to 􀀢 § If !!"\# = 􀀢 (􀀤) then 􀀦 = ∫ 􀀢(􀀤) 􀀩􀀤 The integral sign The expression we want to integrate goes between the integral sign and the 􀀡􀀢 . This is known as the integrand. Indefinite integrals In this chapter, you will only learn how to integrate functions of the form 􀀢!, where 􀀣 is a constant and 􀀣 ≠ −1 . To integrate functions of this form, you can use the following result: § ∫ 􀀢! 􀀡􀀢 = "!!"\#\#$ + 􀀪 The " + 􀀪" is known as the constant of integration. To see why we need to add this constant to our result, consider the following three functions: - 􀀬 = 􀀢 % + 2 - 􀀬 = 􀀢 % - 􀀬 = 􀀢 % − 9 If we differentiate the above functions, the result is the same: &&'" = 2􀀢 because the constant term disappears upon differentiating. But since integration is the reverse of differentiation, we should be able to integrate 2􀀢 and get back to whichever of those functions we started with. To allow for this, we must add the unknown constant of integration, 􀀪, to our end result. The above process is known as indefinite integration. Definite integrals A definite integral is one where the integral is bounded between two limits. The main difference between a definite integral and an indefinite integral is that a definite integral will yield a numerical value, while an indefinite integral will yield a function. To calculate a definite integral: § ∫*) 􀀯 ((􀀢) 􀀲􀀢 = [􀀯(􀀢)]*) = 􀀯 (􀀵) − 􀀯(􀀶) Example 1: Evaluate ∫%+ 􀀢 % 􀀲􀀢 . 7%+􀀢 % 􀀲􀀢 = 8􀀤33932 = The limits of the integral are 2 and 3, written on the top and bottom of the integral sign. We write the result in square brackets, with the limits outside :333; − :233; = 9 − 83 = ~~ ~~193 We substitute our limits into the result, and simplify, Integrating polynomials You can use the following two properties of integrals to integrate expressions where there are multiple terms of the form 􀀰􀀤& § ∫ 􀀱􀀤& 􀀩􀀤 = 􀀱∫ 􀀤 & 􀀩􀀤 = 􀀱 2􀀬􀀭113 + 􀀪 for any constant 􀀼 . § ∫ 􀀤 & + 􀀤' 􀀩􀀤 = ∫ 􀀤 & 􀀩􀀤 + ∫ 􀀤 ' 􀀩􀀤 Here is an example showing how we use these properties in practice: Example 2: Evaluate ∫ 2􀀢+ − 2") + 4􀀢 􀀲􀀢 We can split up the integral as follows: ⇒ 2∫ 􀀢 + 􀀲􀀢 + (−4)∫ 􀀢 1% 􀀲􀀢 + 4∫ 􀀢 􀀲􀀢 ⇒ 2 8􀀢429 + 􀀪 − 4 81$19 + 􀀲 + 4 8􀀢2%9 + 􀁉 ⇒ 􀀢22 + 4􀀢 + 2􀀢% + 􀀼 Each of the integrals yields an integration constant (􀀪, 􀀲 and 􀁉), so we can denote their total as one constant, 􀀼 . Finding the constant of integration You can find the constant of integration if you are given a point the curve passes through or the value of the function at a given point. Example 3 : The curve 􀁊 with equation 􀁋 = 􀀿 (􀀢) passes through the point (4,5). Given that 􀀿& (􀀢) = '!~~√~~(~~'~~%' , find the equation of 􀁊 . 􀀿 & (􀀢) = 􀀢􀀢$"~~\#~~ − 4􀀢$ . \# = 􀀢 ~~ "~~! − 4􀀢 \# ! 􀀢 " 4􀀢" 2 \# 8 ! ∴ 􀀿(􀀢) = 52 − 32 + 􀁌 ⇒ 􀁋 = 5 􀀢 " − 3 􀀢 " + 􀁌 But we are told that at 􀀢 = 4, 􀁋 = 5. Substituting these values: 2 \# 8 ! 5 = 5 (4)" − 3 (4)" + 􀁌 64 64 203 2 \# 8 ! 203 5 = 5 − 3 + 􀁌 ⟹ 􀁌 = 15 ∴ 􀁋 = 5 􀀢 " − 3 􀀢 " + 15 Finding Areas You need to be able to use definite integration to find the area bounded between a curve and the 􀀢 -axis. § The area between a curve 􀀬 = 􀀯 (􀀢), the lines 􀀢 = 􀀶, 􀀢 = 􀀵 and the 􀀢-axis is given by 􀁆􀁊􀁉􀀶 = 7)*􀀯 (􀀢) 􀀲􀀢 Areas under the 􀀵-axis When integrating over an interval where the curve is below the 􀀢-axis, the resultant area will be negative. As a result, you need to take extra care when finding areas under curves which are not strictly positive. § When integrating over an interval where the curve is both above and below the 􀀢-axis, you should split the integral up into separate regions where the function is strictly posit

FAQs

What does the integral symbol represent, and what is the integrand?

The integral symbol represents integration. The expression between the integral sign and the differential is known as the integrand.

How do indefinite and definite integrals differ in their results?

An indefinite integral includes an added constant of integration and yields a function, while a definite integral is bounded by two limits and yields a numerical value.

Why must areas be split when a curve crosses the x-axis?

When the curve is below the x-axis, the integral gives a negative area. If you combine positive and negative regions without splitting, they can cancel out and produce an incorrect total area.

Oct 13, 2026

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