Download Chapter 5 Calculus MAT220

Chapter 5 Calculus MAT220
Mr. Schultz Fall 2005’
NAME_________________________________
Evaluate the following integrals:
#1.
 23 2

3 2 2

x



sin(
x
)
d
x

x



sin
x


dx

 
  x
x


3 53
 x  2ln x   cos( x)  C
5
#2.
x e
2 x3 1
dx
let u  x3  1 then du  3x 2 dx
1 u
e du
3
1
 eu  C
3
1 3
 e x 1  C
3

#3.
sec 2 x
 x dx
let u  x then du 
1
2 x
dx
 2  sec 2 (u ) du
 2 tan(u )  C
 2 tan( x )  C
#4.
x
2
x  1dx let u  x  1 then du  dx and x  u  1 from which x 2   u  1
   u  1
2
udu    u 2  2u  1 udu
3
1
 52

2
   u  2u  u 2 du


2 72 4 52 2 32
 u  u  u C
7
5
3
7
5
3
2
4
2
2
2
  x  1   x  1   x  1 2  C
7
5
3
2
#5
3

x  1dx 
0
1
3
0
1
  x  1 dx    x  1 dx
1
3
 x2

 x2

   x    x
 2
0  2
1
.
1 
9
 1 
   1   0  0     3     1
2 
2
 2 
1 3 1
 
2 2 2
 2.5

4
2
-5
5
-2
-4
#6.
1
cos x sin xdx
1  x2
1

f ( x) 
cos   x  sin   x 
 f ( x) 
1  x
2

cos x sin xdx
1  x2
cos x sin x
 f ( x) Thus, the integrand is an odd function.
1  x2
1
cos x sin xdx
 0 by symmetry
2
1

x
1

#7. Find the exact value of

6
sin  cos
 1  sin
1
2

d Consider the identity 1  sin 2 x  cos 2 x then



6
6
sin 
sin  cos
sin  cos


d


d
1 cos d now let u  cos , du   sin  d
1 cos 2 
1 1  sin 2 
6

sin 
1 cos d  
6
 ln
cos( / 6)

cos(1)
1
du  -ln u
u
cos( /6)
cos(1)
  ln cos  / 6   ln cos(1)
cos(1)
cos( / 6)
This can clean-up to;
ln
2 3 cos(1)
cos(1)
 0.472
 ln
3
3
2
> restart:with(plots):
Warning, the name changecoords has been redefined
> int(tan(x),x=1...Pi/6.0);
>
-0.4717854342
**Note: You could let u be the entire denominator and solve 1 step!
#8.
 (x 
sin(2 x  3))( x  sin(2 x  3))dx    x 2  sin  2 x  3 dx
  x 2 dx   sin  2 x  3 dx
x3 cos  2 x  3
 
C
3
2
#9. Compute
d
dx
e2 x
1
t 2  ln tdt 
e 
2x 2
 ln e2 x 2e2 x
 2e2 x e4 x  2 x
#10. It is 10:00 A.M. and five ants have already entered Melissa and Ah Mihn’s picnic
basket. Ants are notorious followers, so ants from all over the vicinity follow their
brethren into the basket. The culinary treat awaiting them is unsurpassed elsewhere, so
once an ant enters the basket he does not leave. If the rate at which the ants are climbing
into the basket is well modeled by Ants(t) = 100e-0.2t ants per hour, where t = 0 is the
benchmark hour of 10:00 A.M., then answer the following.
A. Write an integral expression to find the number of ants there will be in the basket
x hours after 10:00 A.M..
x
100e
0.2 t
dt  5
0
B. If the girl’s dig into the basket at 1:00 P.M., how many ants will be inside? Give
an exact answer.
3
100 u
e du  5  500 e 0.6  1  5  505  500e0.6  231ants

0.2 0