viii Contents
31.4 Differentiation of further logarithmic
functions
326
31.5 Differentiation of [ f (x)] x
328
Revision Test 9
330
32 Differentiation of hyperbolic functions
331
32.1 Standard differential coefficients of
hyperbolic functions
331
32.2 Further worked problems on
differentiation of hyperbolic functions
332
33 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
341
34 Partial differentiation
345
34.1 Introduction to partial derivatives
345
34.2 First order partial derivatives
345
34.3 Second order partial derivatives
348
35 Total differential, rates of change and small
changes
351
35.1 Total differential
351
35.2 Rates of change
352
35.3 Small changes
354
36 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
361
Revision Test 10
367
37 Standard integration
368
37.1 The process of integration
368
37.2 The general solution of integrals of the
form ax n
368
37.3 Standard integrals
369
37.4 Definite integrals
372
38 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
383
39 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
395
Revision Test 11
397
40 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 x and 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
sin θ substitution
402
40.6 Worked problems on integration using
tan θ substitution
404
40.7 Worked problems on integration using the
sinh θ substitution
404
40.8 Worked problems on integration using the
cosh θ substitution
406
41 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
412
42 The t = tan
θ
2
substitution
414
42.1 Introduction
414
42.2 Worked problems on the t = tan
θ
2
substitution
415
31.4 Differentiation of further logarithmic
functions
326
31.5 Differentiation of [ f (x)] x
328
Revision Test 9
330
32 Differentiation of hyperbolic functions
331
32.1 Standard differential coefficients of
hyperbolic functions
331
32.2 Further worked problems on
differentiation of hyperbolic functions
332
33 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
341
34 Partial differentiation
345
34.1 Introduction to partial derivatives
345
34.2 First order partial derivatives
345
34.3 Second order partial derivatives
348
35 Total differential, rates of change and small
changes
351
35.1 Total differential
351
35.2 Rates of change
352
35.3 Small changes
354
36 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
361
Revision Test 10
367
37 Standard integration
368
37.1 The process of integration
368
37.2 The general solution of integrals of the
form ax n
368
37.3 Standard integrals
369
37.4 Definite integrals
372
38 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
383
39 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
395
Revision Test 11
397
40 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 x and 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
sin θ substitution
402
40.6 Worked problems on integration using
tan θ substitution
404
40.7 Worked problems on integration using the
sinh θ substitution
404
40.8 Worked problems on integration using the
cosh θ substitution
406
41 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
412
42 The t = tan
θ
2
substitution
414
42.1 Introduction
414
42.2 Worked problems on the t = tan
θ
2
substitution
415