7.4, #9. Set t2sint = uv¢, where u = t2, v¢ = sint. Then u¢ = 2t, v = -cost, and

ó
õ 		t2sint dt = -t2cost+ ó
õ 		2tcost dt
The second integral is 2tsint+2cost+C (See example 2 on page 360 of the text). The final answer is -t2cost+2tsint+2cost+C.


7.4, #14. Set sin2 q = uv¢, where u = sinq, v¢ = sinq. Then u¢ = cosq, v = -cosq, and

ó
õ 		sin2q dq = -sinqcosq+ ó
õ 		cos2q dq = -sinqcosq+ ó
õ 		(1-sin2q) dq
using the identity sin2q+cos2q = 1. Thus 2òsin2q dq = -sinqcosq+ò dq. The final answer is (1/2)(-sinqcosq+q)+C.

This problem is similiar to the integral of cos2q which is done in Example 6 of page 363. Another approach is to use the trig identites:

sin2q = 1
2
 (1-cos(2q))       and      sin(2q) = 2sinqcosq
The integral becomes
1
2
 ó
õ 		(1-cos(2q)) dq = 1
2
 q- 1
4
 sin(2q) = 1
2
 (q-sinqcosq).


7.4, #17. Set (lnx)/x2 = uv¢, where u = lnx, v¢ = x-2. Then u¢ = 1/x, v = -x-1, and

ó
õ 		lnx
x2
  dx = - lnx
x
 + ó
õ 		1
x2
  dx = - lnx
x
 - 1
x
 +C.


7.4, #31. Set exsinx = uv¢, where u = sinx, v¢ = ex. Then u¢ = cosx, v = ex, and

ó
õ 		exsinx dx = exsinx- ó
õ 		excosx dx.
Do the second integral in a similar way, that is, view excosx as uv¢, where u = cosx, v¢ = ex. Then u¢ = -sinx, v = ex, and
ó
õ 		excosx dx = excosx+ ó
õ 		exsinx dx.
Plug this into the first answer to get
ó
õ 		exsinx dx = exsinx-(excosx+ ó
õ 		exsinx dx).
Move the integral on the right-side to the left-side and divide by 2 to get the final answer (1/2)ex(sinx-cosx)+C.


7.5, #6. Use formula 12 on page 366 with x = q, ax = 3q, and bx = 5q. Then

ó
õ 		sin3qcos5q dq = 1
16
 (5sin3qsin5q+3cos3qcos5q)+C.


7.5, #8. Use formula 9 on page 366 with a = -3 and b = 1. Then

ó
õ 		e-3qcosq dq = 1
10
 e-3q(-3cosq+sinq)+C.


7.5, #13. Use formula 24 on page 367 with y = x and a = Ö3. Then

ó
õ 		1
3+y2
  dy = 1
Ö3
 arctan y
Ö3
 +C.


7.5, #14. Use formula 14 on page 366 with p(x) = x2 and a = 3. Then

ó
õ 		x2e3x dx = 1
3
 x2e3x- 1
9
 2xe3x + 1
27
 2e3x+C = e3x
27
 (9x2-6x+2)+C.

File translated from TEX by TTH, version 2.00.
On 15 Aug 1999, 15:29.