Chapter 4
Mathematics
CHAPTER 4 Calculus Indeterminate form ,
,
,
,
,
,
, -
, 0.
Example: Plot y= [x] Here [x]
greatest integer not greater than x -2 x < -1 , y = -2 -1 x < 0 , y = - 1 0 x < 1 , y =0 1 x < 2 , y = 2
Various Plots
y
y
Y =ax 0 0 minimum
Maximum value
@x=2
= 16 – 60 +72 +11 = 39
Minimum value
@ x =3
= 54 – 135 +108 +11 =173 – 135 = 38
Example Show that maximum value of
is less than its minimum value.
Solution y= =
,
x=±1 =0+
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Chapter 4
At x = 1,
, =2
At x = -1,
Mathematics
minimum
= -2
maximum
Maximum value @ x = -1 = -1 +
= -2
Minimum value @ x= +1 =
= 2
1+
Example Find the maxima and minima of 5
5
Solution 5
5
5
5
x =0, 1, 3 6 |
= - 10
Maximum value = 1 – 5 + 5 – 1 = 0 |
= 90 > 0
Minimum value = |
+ 5 . 34 + 5 . 33 – 1 = -28
=0 6 |
= 30
Hence neither maxima, nor minima at point x = 0
Example Find the maxima and minima of 6
, in interval [-1, 1]
Solution 6
6
, √
x= Hence
=
= 1 +i : Monotonous function
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Chapter 4
Mathematics
Integration Reverse process of differentiation or the process of summation ∫ any continuous function.
defines the integral of
Standard Integral results 1. ∫ x dx
, n
2. ∫ dx
og x
3. ∫ e dx = e 4. ∫ a dx =
(prove it )
5. ∫ cos x dx
sin x
6. ∫ sin x dx
cos x
7. ∫ sec x dx
tan x
8. ∫ cosec x dx
cot x
9. ∫ sec x tan x dx
sec x
10. ∫ cosec x cot x dx 11. ∫ √ 12. ∫ 13. ∫
√ √
dx dx dx
cosec x
sin sec sec
x
14. ∫ cosh x dx
sinh x
15. ∫ sinh x dx
cosh x
16. ∫ sech x dx
tanh x
17. ∫ cosech dx
coth x
18. ∫ sech x tanh x dx
sech x
19. ∫ cosech x cot h x dx
cosech x
20. ∫ tan x dx
og sec x
21. ∫ cot x dx
og sin x
22. ∫ sec x dx
og sec x
23. ∫ cosec x dx
tan x = og tan ⁄
og cosec x
⁄
cot x = log tan
24. ∫ √
dx
og x
√x
a
= cosh
25. ∫ √
dx
og x
√x
a
= sinh
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Chapter 4 √
26. ∫ √a
x dx
sin
27. ∫ √a
x dx
√x
a
og x
√x
a
28. ∫ √x
a dx
√x
a
og x
√x
a
29. ∫
dx = tan
30. ∫
dx =
31. ∫
dx =
og
where x a
sin x
33. ∫ cos x dx
sin x
34. ∫ tan x dx
tan x
35. ∫ cot x dx
x
cot x
36. ∫ n x dx
x nx
x x
37. ∫ e
sin bx dx
a sin bx
b cos bx
38. ∫ e
cos bx dx
a cos bx
b sin bx
39. ∫ e [f x
Mathematics
f x ]dx
e f x
Method of finding Integrals: (A) (B) (C) (D)
Integration by INSPECTION Integration by TRANSFORMATION Integration by SUBSTITUTION Integration by PARTS
Integration by parts:
∫ u v dx
u. ∫ v dx
∫
∫ v dx dx
I L A T E E
Selection of U & V Inverse circular (e.g. tan
x) Logarithmic
Exponential Algebraic Trigonometric
Note: Take that function as “u” which comes first in “ILATE” THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter 4
Mathematics
Some Other Important Formulae Area = ∫ y dx = Vo u e
∫ r d
π ∫ y dx=
∫ r sin d
cos x
sin x
cos x
cos x
sin x
sin x
sin x
cos x
cos x
cos x
Example ∫ sec x tan x dx = ? Solution = x =
x
∫ sec x tan x dx
∫ sec
tan
=
=
sec
Example ∫ sin
x dx
∫ . sin
= ∫ sin x . dx = sin x, x ∫ √ = x sin
x
∫
= x sin
x
+ .
= x sin
x
x dx
. x dx /
√
. x dx
(z = 1 – x , dz = -2x dx )
/
/
x
/
Rules for definite integral 1)∫
=∫
2) ∫
=∫
+∫
a
