INTEGRAL CALCULUS dy Please use of the following as guide in solving the assignment. Antidifferentiation Differential Equation Chain Rule Logarithmic Funtion Exponential Function Integration by Parts...

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Answered Same DayDec 23, 2021

Answer To: INTEGRAL CALCULUS dy Please use of the following as guide in solving the assignment....

Robert answered on Dec 23 2021
121 Votes
Sol: (1)
 
3
5
41 2
y
dy
y

Substitute
4
3
1 2
8
u y
du y dy
 
 

Now,
   
 
 
 
   
3
5 5
4
3
5
5
4
3 5 1
5
4
3 4
5
4
3
5 4
4 4
1 1
81 2
1
81 2
1

8 5 11 2
1
8 41 2
1
1 2 32 1 2
y
dy du
uy
y
dy u du
y
y u
dy C
y
y u
dy C
y
y
dy C
y y

 

 

 

 
   
  
 
   
 
 
 
 
 



Sol: (2)  2sec 5x dx
 2 int sec 5 , 5 5 :For the egrand x substitute u x and du dx 
   
 
 
2 21sec 5 sec
5
1
tan
5
1
tan 5
5
x dx u du
u C
x C

 
 
 

Sol: (3)
1
3 2
dx
x


1
int , 3 - 2 -2 :
3 2
For the egrand substitute u x and du dx
x
 


 
 
1 1 1
3 2 2
1
log
2
1
log 3 2
2
dx du
x u
u C
x C
 

  
   
 
Sol: (4)
sin 3
cos3 1
t
dt
t 

sin 3
int , cos3 1 3sin 3 :
cos3 1
t
For the egrand substitute u t and du tdt
t
   


 
 
sin 3 1 1
cos3 1 3
1
log
3
1
log cos3 1
3
t
dt dt
t u
u C
t C
 

  
   
 

Or

sin 3 2 3
log sin
cos3 1 3 2
t t
dt C
t
  
    
   

Sol: (5)
21 x
x
e
dx
e


2 2
2
1 1
1
1
x x
x x x
x x
x x
x
x
x
x
e e
dx dx dx
e e e
e dx e dx
e e C
e C
e
e
C
e



 
 
   
   

 
  
 
Sol: (6)
 
 
3
3
cos 3
sin 3
x
dx
x

 
 
      
 
3
3
2
3 3
2
cos 3
int ,
sin 3
sin 3 sin 3 cos 3
cos 3

x
For the egrand
x
substitute u x and du x x dx
x
du dx
u

 


 
 
    
 
 
 
 
     
23
3 3
6
2
6
7
2 8
2 8
3 3
cos 3 1 sin 3cos 3
sin 3 sin 3
1
1
2 8
sin 3 sin 3
2 8
x xx
dx dx
x x
u
u du
u
u udu
u u du
u u
C
x x
C




 
 
   
     
   
   
   
     
   
   
 



Sol: (7)
5 3tan secx xdx
5 3 2 2 For the integrand tan sec , use the triginometric identity tan sec 1:x x x x 
      
2
5 3 3...
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