viii Contents
31.4 Differentiation of further logarithmic
functions 326
31.5 Differentiation of [f(x)]x 328Revision Test 9 33032 Differentiation of hyperbolic functions 331
32.1 Standard differential coefficients of
hyperbolic functions 331
32.2 Further worked problems on
differentiation of hyperbolic functions 33233 Differentiation of inverse trigonometric and
hyperbolic functions 334
33.1 Inverse functions 334
33.2 Differentiation of inverse trigonometric
functions 334
33.3 Logarithmic forms of the inverse
hyperbolic functions 339
33.4 Differentiation of inverse hyperbolic
functions 34134 Partial differentiation 345
34.1 Introduction to partial derivatives 345
34.2 First order partial derivatives 345
34.3 Second order partial derivatives 34835 Total differential, rates of change and small
changes 351
35.1 Total differential 351
35.2 Rates of change 352
35.3 Small changes 35436 Maxima, minima and saddle points for functions
of two variables 357
36.1 Functions of two independent variables 357
36.2 Maxima, minima and saddle points 358
36.3 Procedure to determine maxima, minima
and saddle points for functions of two
variables 359
36.4 Worked problems on maxima, minima
and saddle points for functions of two
variables 359
36.5 Further worked problems on maxima,
minima and saddle points for functions of
two variables 361Revision Test 10 36737 Standard integration 368
37.1 The process of integration 368
37.2 The general solution of integrals of the
formaxn 368
37.3 Standard integrals 369
37.4 Definite integrals 37238 Some applications of integration 375
38.1 Introduction 375
38.2 Areas under and between curves 375
38.3 Mean and r.m.s. values 377
38.4 Volumes of solids of revolution 378
38.5 Centroids 380
38.6 Theorem of Pappus 381
38.7 Second moments of area of regular
sections 38339 Integration using algebraic substitutions 392
39.1 Introduction 392
39.2 Algebraic substitutions 392
39.3 Worked problems on integration using
algebraic substitutions 392
39.4 Further worked problems on integration
using algebraic substitutions 394
39.5 Change of limits 395Revision Test 11 39740 Integration using trigonometric and hyperbolic
substitutions 398
40.1 Introduction 398
40.2 Worked problems on integration of sin^2 x,
cos^2 x,tan^2 xand cot^2 x 398
40.3 Worked problems on powers of sines and
cosines 400
40.4 Worked problems on integration of
products of sines and cosines 401
40.5 Worked problems on integration using the
sinθsubstitution 402
40.6 Worked problems on integration using
tanθsubstitution 404
40.7 Worked problems on integration using the
sinhθsubstitution 404
40.8 Worked problems on integration using the
coshθsubstitution 40641 Integration using partial fractions 409
41.1 Introduction 409
41.2 Worked problems on integration using
partial fractions with linear factors 409
41.3 Worked problems on integration using
partial fractions with repeated linear
factors 411
41.4 Worked problems on integration using
partial fractions with quadratic factors 41242 Thet=tan
θ
2
substitution 414
42.1 Introduction 414
42.2 Worked problems on thet=tanθ
2
substitution 415