Page 67 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 1 1

x
x
 (sinx  cos )  ( cosx  sin )
 I   2 2 dx
0 sinx  cosx

1  1  cosx  sinx
  1dx   dx
2 2 sinx  cosx
0 0
 
1  1 
 x  log|sinx  cos |
x
2  0  2  0 
1 1

 |  0| log|sin  cos | log|sin0 cos0| 


2 2
 1 
  log| 1| log|1|   
2 2 2
Example: Evaluate the following integrals:
1
x
1.  xe dx
0
2
2.  logxdx
1
 /2
 /2   /2
3.  x cosxdx  sinx    sinxdx
0  0  0

x
4.  cos2 logsinxdx
0
 /6
2
x

5.  (2 3 )cos3xdx
0
4 x  x
2
6.  dx
2 2x  1
e x  1 x logx 

7.  e  x  dx
1  
Solution:
1
x
1.  xe dx
0
1
1 x
x
 xe   e dx
0  
0
1
x
0
1
]

 [1.e  0] e    [e e  e e  1 1


e

0 
2
2.  logxdx
1
2
2 1
1 
 log .x   .xdx
x
1 x
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