What is the integration of 1 sinx + secx dx

ANTIDIFFERENTIATION

AND
INDEFINITE INTEGRAL
CONTENTS
• Antidifferentiation and the Indefinite Integral
• Basic Theorems on Antidifferentiation
• The Chain Rule for Antidifferentiation
• The definite integral and area of ​​a plane
• Theorems on Definite Integral
• Area of ​​a plane region
ANTIDIFFERENTIATION
• It is also called integration.
• It is the reverse process to differentiation.
• It is the inverse operation of differentiation.
• It involves the computation of an antiderivative.
DEFINITION
A function F is called
antiderivative of the function f
on an interval I if
F ' x  f  x
for every value of x in I.
Example
If F is the function defined by
F , x then
 4x  x  5
3 2
F ' xif f 12
Thus 2
is xthe 2x
function defined by
f  x   12 x  2 x, then f is the derivative of
2

F and F is an antiderivative of f.
ANTIDIFFERENTIATION
The symbol ∫ denotes the operation
of antidifferentiation and we write,
where  f  x dx  F  x  C
F ' x   f  x and d  F  x    f  x the
dx
expression F  x  is C the general
antiderivative of f.
THEOREMS ON ANTIDERIVATIVES
1.  dx  x  C

2.  a f  x   a  f  x  dx where a is a constant.

3. If f and g are defined on the same interval,


then
  f  x   g  x     f  x  dx   g  x  dx
THEOREMS ON ANTIDERIVATIVES
4. If f1, f2, ...., fn are defined on the same interval,

  c f  x   c f  x   .....  c f  x   dx 
1 1 2 2 n n

c  f  x  dx  c  f  x  dx  ...  c  f  x  dx
1 1 2 2 n n

where c1, c2, ...., cn are constants.


n 1
x
 x dx  n  1  C; n  1
n
5. If n is a rational number,
Example: Evaluate the following.

1 .  x dx
2
2.   3 x  5  dx

3.   5 x 4
 8 x  9 x  2 x  7 dx
3 2

 1 5t  7
2
4.  x  x  dx
 x
5.  4
dx
t 3
n 1
x
1 .  x dx
2
by Theorem 5
 xn
dx   C; n  1
n 1
2 1
x
  
2
x dx C
2 1
3
x
 C
3
1 3
 x C
3
2.   3 x  5  dx by Theorem 3

  f  x   g  x     f  x  dx   g  x  dx

  3 x  5  dx   3 x dx   5 dx
 3 x dx  5 dx by Theorem 2
2
3x
  5x  C
2
3.   5 x 4
 8 x  9 x  2 x  7 dx
3 2

  5 x dx   8 x dx   9 x dx   2 x dx   7dx
4 3 2

5 4 3 2
x x x x
 5  8  9   2   7x  C
5 4 3 2
 x  2 x  3x  x  7 x  C
5 4 3 2
1
 1   x  x  x 1  dx
 x  x  dx 
2
4.
 x 3

1
  x 2  x 2 dx
3 1

  x dx   x dx
2 2

5 1
x 2
x 2
  C
5 1
2 2
5 1
2
 x  2x  C
2 2
5
5t  7
2 2 4



5. 4
dt   5t  7t
3 3
German
t 3 2
  5t dx   7t
3

4
3
German
5 1

t
3
t 3
 5  7 C
5 1
3 3
5
21
 3t  1  C
3

t3
SEATWORK: Evaluate the following:
1 .  3 x dx
4

2
2.  3 dx
x


3.  y 2 y  3 dy
3 2

THEOREM ON ANTIDERIVATIVES OF TRIGONOMETRIC FUNCTIONS

1.sinx dx cosx C

2. cos x dx  sin x  C


3. sec x dx  tan x  C
2

4. csc x dx   cot x  C


2

5. sec x tan x dx  sec x  C


6. csc x cot x dx   csc x  C
TRIGONOMETRIC IDENTITIES
sin x csc x  1 sin x cos x
tan x  cot x 
cos x  sec x  1 cos x sin x
tan x  cot x  1 sin x  cos x  1
2 2

tan x  1  sec x
2 2

cot x  1  csc x
2 2
EXAMPLES: Evaluate the following:

2 cot x  sin x
2
1. dx
sin x

2.  tan x  cot x  4 dx
2 2

 
3.  3 sec x tan x  5 csc x dx
2
2 cot x  3 sin x   2 cot x  3 sin x  dx
2 2

1.  dx   
sin x  sin x sin x 
 1 
 2   cot x   dx  3 sin x dx
 sin x 
 2  cot x csc x dx  3 sin x dx

 2 csc x  3 cos x  C
 
2.  tan x  cot x  4 dx
2 2

  tan x dx   cot x dx   4 dx
2 2

 2
  2

  sec x  1 dx   csc x  1 dx   4 dx

  sec x dx   dx   csc x dx   dx   4 dx
2 2

 tan x  x  cot x  x  4 x  C
 tan x  cot x  2 x  C

3.  3 sec x tan x  5 csc x dx
2

  3 sec x tan x dx   5 csc x dx
2

 3 sec x  5 cot x  C
SEATWORK: Evaluate the following:

sin x
1.  2
dx
cos x


2.  4 csc x cot x  2 sec x dx
2

 
3.  2 cot x  3 tan x dx
2 2