The Chemistry Maths Book, Second Edition

6.8 Exercises 187

6.8 Exercises

Section 6.2

Evaluate the indefinite integrals:

1. 
   
   
   
   2. 
      
      
      
      3. 
         
      
      
      
      
   
   
   
   


2. 
   
   
   
   5. 
      
      
      
      6. 
         
      
      
      
      
   
   
   
   

Evaluate the definite integrals:

7. 
   
   
   
   8. 
      
      
      
      9. 
         
      
      
      
      
   
   
   
   

10.The wave functions for a particle in a box of length lare

Show that the functions satisfy the orthonormality conditions

Section 6.3

Evaluate the indefinite integrals (use the substitutions in parentheses, when given):

11. (u 1 = 13 x 1 + 1 1) 12.


12. (u 1 = 13 x

2

1 + 12 x 1 + 1 5) 14.

15. (u 1 = 1 x

3

1 + 12 x) 16.

17. (u 1 = 141 − 1 x

2

) 18.

19. (u 1 = 111 + 1 e

x

) 20.

21. (u 1 = 1 x

2

1 + 1 x 1 + 1 2) 22.

23. (u 1 = 111 − 1 sin 1 x) 24.


24. 
    
    
    
    26. Z
    
    
    
    

tan

ln(cos )

x

x

dx

Z

x

x

dx

4

2

−

Ztan xdx

Z

cos

sin

x

x

dx

1 −

Z

3

23

2

32

xx

xx

dx

−

−+

Z

21

2

2

x

xx

dx

• 
  

++

Zxxdx cos(3 )

2

Zeedx − 1

xx

()1

12

• 
  

Zcos

sin

xe dx

2 x

Zxxdx 4

2

−

Z()1

42

2

−

−

xe dx

xx

Z()

()

32

22

3

xe dx

xx

• 
  

−+

Z()()23121

3132

xx x dx+− +

Z()()32531

23

xx xdx++ +

Z()21

12

Z()31 xdx−

5

xd+ x

Z

0

1

0

l

nm

dx

mn

mn

ψψ =

=

≠











if

if

ψ

n

x

l

nx

l

() sin= n













,=,,,

2

123

π

...

Z

0

2

π

Z sinxxdxcos

0

2

2

π

Z sin 2xxdxcos

0

2

3

π

cos

2

xdx

Zcos 5 cos 2xxdx

Zsin 3xxdxsin

Zsin cosxxdx 3

Zsin Zsin 3xxdxcos 3 Zsin 3xxdxcos 2

2

3 xdx