7.2, #12. Set w = -x2, dw = -2x dx, then

ó
õ 		xe-x2 dx = - 1
2
 ó
õ 		ew dw = - 1
2
 ew+C = - 1
2
 e-x2+C.



7.2, #16. Set w = y+5, dw = dy, then

ó
õ 		1
y+5
  dx = ó
õ 		1
w
  dw = lnw = ln(y+5).


7.2, #20. Set w = 4-x, dw = -dx, then

ó
õ 		dx
  ___
Ö4-x
 
= - ó
õ 		w-1/2 dw = -2w1/2+C = -2   ___
Ö4-x
 
 +C.


7.2, #24. Set w = cos3t, dw = -3sin3t dt, then

ó
õ 		  æ
Ö
cos3t
 
 sin3t dt = - 1
3
 ó
õ 		w1/2 dw = - 2
9
 w3/2+C = - 2
9
 (cos3/23t)+C.


7.2, #28. Set w = Öx, dw = 1/(2Öx) dx, then

ó
õ 		cosÖx
Öx
  dx = 2 ó
õ 		cosw dw = 2sinw+C = 2sinÖx+C.


7.3, #1(a). Set w = 1+x2, dw = 2x dx. Then w = 1 if x = 0, w = 2 if x = 1, and

ó
õ 		1

0 
 x
1+x2
  dx = 1
2
 ó
õ 		2

1 
 dw
w
 = 1
2
 ln|w| ê
ê
ê 		2

1 
 = 1
2
 ln2 = 0.347

#1(b). Set w = cosx, dw = -sinx dx. Then w = 1 if x = 0, w = 1/Ö2 if x = p/4, and

ó
õ 		p/4

0 
 sinx
cosx
  dx = - ó
õ 		1/Ö2

1 
 dw
w
 = -ln|w| ê
ê
ê 		1/Ö2

1 
 = -ln(1/Ö2) = 0.347.
The answers are the same since -ln(1/Ö2) = lnÖ2 = (1/2)ln2.


7.3, #2. Set w = px, dw = pdx. Then w = 0 if x = 0, w = p/2 if x = 1/2, and

ó
õ 		1/2

0 
 cospx dx = 1
p
 ó
õ 		p/2

0 
 cosw dw = 1
p
 sinw ê
ê
ê 		p/2

0 
 = 1
p
 = 0.318.


7.3, #3. Set w = x1/3, dw = (1/3)x-2/3 dx. Then w = 1 if x = 1, w = 2 if x = 8, and

ó
õ 		8

1 
 e3 Öx
   __
3Öx2
 
 dx = 3 ó
õ 		2

1 
 ew dw = 3ew ê
ê
ê 		2

1 
 = 3(e2-e) = 14.012.

File translated from TEX by TTH, version 2.00.
On 14 Aug 1999, 21:21.