; Integration
Documents
User Generated
Resources
Learning Center
Your Federal Quarterly Tax Payments are due April 15th

# Integration

VIEWS: 3 PAGES: 22

• pg 1
```									Integration
Integration (Anti-derivative)
∫2x dx =» What is the function whose derivative is 2x
Example 1 : Find the indicated integral


 3x   2

 5x  6 dx
Solution

= x3 
5x 2
 6x  C          PLUS A CONSTANT
2
Example 2: Find the indicated integral

 1    

  x2

dx

Solution

 x dx
2
=

1
=  C             PLUS A CONSTANT
x
Example 3: Find the indicated integral


 x  1x  1dx
Solution

 x         
1 dx
2
=

x3
=     xC             PLUS A CONSTANT
3
How to integrate common funtions:

In general,

n 1

 
x
x dx 
n
 C ; n  1
n 1
Example 4: Find the indicated integral

x 4  3x 3  4 x 2  1
             2
dx
x

Solution

1
=    x  3x  4  x 2 dx
2

= x 3 3x 2       1
      4x   C
3    2         x
Exponential Rule

In general,

1 kx
 e dx  k e  C
kx
Examples 5,6,&7: Find the indicated integral
 ∫e dx = ex + C

  e x dx =           ex  C

e 2 x 5
e
2 x 5
                 dx =          C
2
The Logarithmic Rule

In general,

1
     x
dx =   ln | x | C
In general,

f ' ( x)
    f ( x)
dx  ln | f ( x) | C
Example 8: Find the indicated integral

2x

   x 1
2
dx

=       
ln x 1  C
2

Example 9: Find the indicated integral

x 1
      x2  2x  5
Solution

1  2x  1
=  2
2 x  2x  5

1
=   ln | x 2  2 x  5 | C
2
Example 10
 If the slope of the tangent of f(x) is 3x 2  5x, find the
equation of the curve of this function which passes
through (1,2).
Solution
f ’(x)= 3x 2  5x
f(x)=∫ 3x 2  5x dx
2
y= x  5 x  C
3

2
Since the curve passes through (1,2), then (1,2) satisfies
its equation.
5
2=1+ 2 +C                  5x 2 3
C=  3              yx 
3

2                      2     2
The Definite Integral
The Fundamental Theorem of Calculus

If the function f(x) is continuous on the interval
b
a ≤ x ≤ b, then    f ( x)dx  F (b)  F (a)
a
where F(x) is any

anti-derivative of f(x) on a ≤ x ≤ b.
Example 11: Evaluate the definite integral

  x 3  3x dx
1

2

Solution
1

=  x  3x    
4     2

           
 4     2 2

1 3
=       4  6 
4 2

1
=   8
4
Example 12: Evaluate the definite integral

 3x               
2
           2
 2 x  1 dx
1

Solution

= x3  x 2  x 2
1        
= 14-13 = 11
Example 13: Find the area under the curve of the
following function

 x             
3
            2
 1 dx
1

Solution

 x             
3
A           2
 1 dx
1

3
 x3    
      x
 3     1

1 
 9  3    1            12 
4
 10.67
3                       3
Example 13 (Graph)

f ( x)  x 2  1
Area Between two Curves

In general, Area between 2 curves f(x) and g(x)

b
A    f ( x)  g ( x)dx
a
Example 14
 Find the area of the region R enclosed by the curves y  x 3
and y  x 2

Solution
To find the points of intersection of the 2 curves, we solve the equations
simultaneously : x 3  x 2 => x 2  x  1  0 => x=1 or x=0
1
A   x  x dx => A 
1
 x 4 x3  1 1                    1    1
   =  4  3             = 
3   2
=
0
4    3 0                       12   12
Example 15
 Find the area of the region enclosed by the line y  4 x and the curve
y  x 3  3x 2

Solution
To find the point of intersection, put x 3  3x 2  4 x

x x 2  3x  4  0 
 x                             x                   
0                               1
A          3
 3 x  4 x dx 
2                        3
 3 x 2  4 x dx
x x  4 x  1  0              4                               0

x  0, x  4, x  1              x4
0                                1
2       x4              
A      x  2x 
3
      x3  2 x 2 
 4            4    4               0

1                         1          
A  0    4  (4)3  2(4) 2     1  2 
4

4                         4          
3                        3
A   32                          A  32
4                        4

```
To top