Chapter 7 ยท Introduction

Chapter 7 โ€” Integrals

Differentiation turns a function into its rate of change. Integration runs that machine backwards โ€” from a speed to a distance, from a slope to a curve, from a rate to a total โ€” and computes areas along the way.

A car's speedometer tells you how fast you are going at each instant; the odometer tells you how far you have gone. If you had only a record of the speedometer โ€” say, a list of speeds every second for a journey from Pune to Mumbai โ€” could you rebuild the odometer reading? A hospital knows the rate at which a drip delivers a drug, in milligrams per minute: how much has the patient received after an hour? A water-tank engineer knows the inflow rate through the day: how much water is in the tank at 6 p.m.? In each case the derivative of a quantity is known and the quantity itself is wanted. In Chapter 5 you learnt to go from a function to its derivative; this chapter goes the other way.

Running differentiation backwards is called integration. The answer to โ€œwhich function has derivative $2x$?โ€ is not one function but a whole family, $x^2+C$, because adding a constant does not change a slope โ€” that is the constant of integration. Unlike differentiation, integration has no rule that always works, so most of the chapter is about methods: a table of standard results read off from known derivatives, substitution (the chain rule backwards), special integrals for quadratic denominators, partial fractions for rational functions, and integration by parts (the product rule backwards). Then a second, quite different question โ€” the area under a curve โ€” turns out to have the same answer. That surprising link is the fundamental theorem of calculus, and it turns every area problem in Chapter 8 into a short calculation.

Derivative knownโ†’Antiderivative + Cโ†’Standard integralsโ†’Substitutionโ†’Special integrals & partial fractionsโ†’By partsโ†’Area = F(b) โˆ’ F(a)โ†’Properties Pโ‚€โ€“Pโ‚†
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • explain what an antiderivative and an indefinite integral are, why every answer carries $+\,C$, and check any answer by differentiating it;
  • write down the standard integrals of powers, exponentials, $\dfrac1x$, trigonometric functions and the inverse-trigonometric forms, and use the linearity of integration;
  • integrate by substitution, including with trigonometric identities such as $\sin^2x=\tfrac12(1-\cos2x)$, and find $\displaystyle\int\tan x\,dx$, $\displaystyle\int\sec x\,dx$ and their cousins;
  • use the special integrals for $\dfrac{1}{x^2\pm a^2}$, $\dfrac{1}{a^2-x^2}$ and $\dfrac{1}{\sqrt{x^2\pm a^2}}$, $\dfrac{1}{\sqrt{a^2-x^2}}$, and reduce any quadratic denominator to them by completing the square;
  • split a rational function into partial fractions (distinct linear, repeated linear and $x^2+a^2$-type factors) and integrate it;
  • integrate by parts โ€” choosing the first function with the ILATE order โ€” including $\displaystyle\int e^x[f(x)+f'(x)]\,dx$ and $\displaystyle\int\sqrt{a^2-x^2}\,dx$;
  • evaluate definite integrals with the fundamental theorem of calculus, change the limits under a substitution, and use the properties Pโ‚€โ€“Pโ‚† (including $\displaystyle\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$ and even/odd functions);
  • answer Board 1- to 5-mark questions and JEE Main/Advanced problems on integration.
Teaser โ€” One Slope, a Whole Family of Curves

Choose a slope function $f$. Every faint curve has slope $f(x)$ at every $x$ โ€” they are the graphs of $F(x)+C$ for different constants $C$, stacked vertically. The red curve is the one with your value of $C$. Play moves the dashed line: at each $x$ the short blue tangent segments on all the curves are parallel, because they all have the same slope $f(x)$. Integration can find the shape โ€” but not the height. That is what $+\,C$ means.

$x$โ€”
slope $f(x)$ (every curve)โ€”
red curve $F(x)+C$โ€”

Sections in this chapter

7.1

Antiderivatives and Integration by Substitution

Integration as the inverse of differentiation; the constant of integration; standard integrals and their properties; substitution and trigonometric identities.

$\int f(g(x))\,g'(x)\,dx=\int f(u)\,du$
Open section โ†’7.2

Integrals of Some Particular Functions and Partial Fractions

The six special integrals; completing the square for quadratic denominators; splitting rational functions into partial fractions.

$\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C$
Open section โ†’7.3

Integration by Parts

The product rule run backwards; choosing the first function by ILATE; $e^x[f+f']$; integrals of $\sqrt{a^2-x^2}$ and $\sqrt{x^2\pm a^2}$.

$\int e^x[f(x)+f'(x)]\,dx=e^xf(x)+C$
Open section โ†’7.4

Definite Integrals, the Fundamental Theorem and Their Properties

Area under a curve; the area function and the two fundamental theorems; substitution with limits; properties Pโ‚€โ€“Pโ‚†.

$\int_a^b f(x)\,dx=F(b)-F(a)$
Open section โ†’
โœ… Before you start
Integration is differentiation read backwards, so you need your derivatives at your fingertips: powers, $e^x$ and $\log x$, the six trigonometric functions, and the chain rule (Chapter 5 โ€” Continuity and Differentiability). The inverse trigonometric functions and their principal values (Section 2.1) appear as answers in Sections 7.2 and 7.3, and their derivatives, such as $\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}$, are also in Chapter 5. From Class 11 Mathematics, Ch. 3 โ€” Trigonometric Functions: the identities $\sin^2x+\cos^2x=1$, $1+\tan^2x=\sec^2x$ and the double-angle formulas; and from Ch. 12 โ€” Limits and Derivatives: the idea of a derivative as a slope. Throughout this chapter, $\log x$ means the natural logarithm $\log_e x$. Quick self-test: differentiate $x^4$, $\sin 3x$, $e^{-2x}$ and $\log(x^2+1)$. (Answers: $4x^3$, $3\cos3x$, $-2e^{-2x}$, $\dfrac{2x}{x^2+1}$ โ€” now read each one backwards: these are four integrals you already know.)
โ† 6.4 Absolute Extrema and Optimisation Problems7.1 Antiderivatives and Integration by Substitution โ†’