MATHS 1

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, Some Useful Formulae
1. sin ix = i sin hx
2. cos ix = cos hx
e ix − e − ix
3. sin x =
2i
e ix + e − ix
4. cos x =
2
1
5. Sin h2x = (cosh 2x – 1)
2
1
6. cos h2x = (cosh 2x + 1)
2
ax
7. z a x dx =
log a
a ≠ 1, a > 0

1
8. z sin haxdx =
a
cos hax

1
9. z cos haxdx =
a
sin hax

1
10. z tan haxdx =
a
log cos hax

1 x x
11. z a2 − x2
dx = sin −1
a
= arc sin
a
1
12. z 2
x −a 2
dx = log x + x 2 − a 2

1 1 x x
13. z 2
x +a 2
dx =
a
tan −1
a
= arc tan
a
x a2 x
14. z a 2 − x 2 dx =
2
a2 − x2 +
2
sin −1
a

e ax
15. z e ax sin bx dx =
a2 + b2
(a sin bx – b cos bx)

e ax
16. z e ax cos bx dx =
a 2 + b2
(a cos bx + b sin bx)

1
17. z sec ax dx =
a
log |sec ax + tan ax|

, 1
18. z cosec ax dx =
a
log |cosec ax – cot ax|

x 3 x5
19. sin x = x − + ......
3 5

x2 x 4
20. cos x = 1 − + ........
2 4

x3 2 5 17 7
21. tan x = x + + x − x + ........
3 15 315
x2 x3 x 4
22. log (1 + x) = x − + − + ........
2 3 4
x2 x3
23. log (1 – x) = − x − − − .........
2 3
x 3 x5
24. sin hx = x + + + .........
3 5

d x
25. a = ax loge a
dx
d 1
26. cos −1 x = −
dx 1 − x2
d 1
27. cot −1 x = −
dx 1 + x2
d 1
28. cosec −1 x = −
dx x x2 − 1
d 1
29. log a x = log a e
dx x
d x a
30. tan −1 = 2 .
dx a a + x2

, Contents
PREFACE TO THE SECOND REVISED EDITION



SOME USEFUL FORMULAE

U NIT I. Differential Calculus-I 1–94
1.0 Introduction 1
1.1 nth Derivative of Some Elementary Functions 1
Exercise 1.1 6
1.2 Leibnitz’s Theorem 7
Exercise 1.2 13
Exercise 1.3 19
Partial Differentiation 20
1.3 Function of Two Variables 20
1.4 Partial Differential Coefficients 21
Exercise 1.4 33
1.5 Homogeneous Function 35
1.6 Euler’s Theorem on Homogeneous Functions 36
Exercise 1.5 47
1.7 Total Differential Coefficient 48
Exercise 1.6 62
Curve Tracing 63
1.8 Procedure for Tracing Curves in Cartesian Form 64
Exercise 1.7 71
1.9 Polar Curves 73
Exercise 1.8 77
1.10 Parametric Curves 78
Exercise 1.9 80
Expansion of Function of Several Variables 81
1.11 Taylor’s Theorem for Functions of Two Variables 81
Exercise 1.10 89
Objective Type Questions 90
Answers to Objective Type Questions 94

U NIT II. Differential Calculus-II 95–150
2.1 Jacobian 95
Exercise 2.1 109