web-7.1 - Web 7.1 In w1w4 integrate by parts w1 x3 ex dx =...

Web 7.1
In w1–w4, integrate by parts.
w1.
Z
x
3
e
x
2
dx =
Z
[ x
2
] e
x
2
x dx
= x
2
1
2
e
x
2
-
Z
1
2
e
x
2
2 x dx by parts
= ” -
1
2
Z
e
x
2
2 x dx
= ” -
1
2
e
x
2
+ C by B3
w2. Do 35 in Ex 7.1 in the book without first substituting.
Solution.
Z √
π
√
π
2
θ
3
cos θ
2
dθ =
Z √
π
√
π
2
[ θ
2
] (cos θ
2
θ dθ )
= θ
2
1
2
sin θ
2
-
Z
1
2
sin θ
2
2 θ dθ
√
π
√
π
2
= · · ·
w3. Do 30 in Ex 7.1 in the book.
Solution.
Z
1
0
r 3
√
4 + r 2
dr =
Z
1
0
[ r
2
] (4 + r
2
)
-
1
2
r dr
= r
2
(4 + r
2
)
1
2
-
Z
(4 + r
2
)
1
2
2 r dr
1
0
= · · ·
w4.
Z
x
3
dx
5
√
9 + x
2
=
Z
[ x
2
] (9 + x
2
)
-
1
5
x dx
= x
2
1
2
5
4
(9 + x
2
)
4
5
-
Z
5
8
(9 + x
2
)
4
5
2 x dx
=
5
8
x
2
(9 + x
2
)
4
5
-
25
72
(9 + x
2
)
9
5
+ C
In w5 and w6, integrate by parts twice. Then transpose the desired unknown integral from
the right side to the left side of the equation. Then solve the equation for this unknown
integral.
w5.
Z
[ e
ax
] (cos bx dx ) = e
ax
1
b
sin bx -
a
b
Z
[ e
ax
] (sin bx dx )

=
1
b
e
ax
sin bx -
a
b
e
ax
- cos bx
b
-
Z
- cos bx
b
e
ax
a dx
!
=
1
b
e
ax
sin bx +
a
b
2
e
ax
cos bx -
a
2
b
2
Z
e
ax
cos bx dx
Now we write the integral at the beginning of this ”string” equal to the expression after
the last ”equal sign” and we get: