4章の演習問題（全問）

(10)
2 2
x asin t, dx acos tdt, a x dx acost acost dt
= = − =
∫
2 2 2 2
a cos tdt = a sin 2t+ t+ C = a sin tcost+
t
2 2 2
⎛1 1 ⎞ ⎛1 1
⎜ ⎟ ⎜
⎝4 2 ⎠ ⎝2 2
a x ⎛ x⎞
a x
= 1− ⎜ ⎟ + arcsin + C
2 a ⎝a⎠
2 a
C ⎞ + ⎟⎠
[2]
(1)
(2) u = (x+3)
(3)
(4)
[1](10)
(5) t = sin x, dt = cos xdx, z = 1−t
(6)
(7)
b
3 b
− ( 32)
+ 1 −1/2
b b
dx ⎡
−3/2
(3 −x) ⎤ ⎡ (3 −x) ⎤ ⎡ 2 ⎤
≡ lim (3 − x) dx= lim lim lim
3/2
⎢− ⎥ = − =
(3 −x) b→3 b→3 ( 3 2)
1
⎢
b→3 (1/ 2)
⎥
b→3⎢ ⎥
0 0 ⎣ − + ⎦0
⎣ − ⎦0
⎣ 3 − x ⎦0
∫ ∫ =∞
x
dx e dx x x dt
(8) = , t = e , dt = e dx→
x − x 2x
2
e e e 1 t 1
+ + + , ∫ [ t]
∞
0
dt
∞ π
= arctan = −0
2 0
t + 1 2
(9) x=1
2 b
2
2
1
2
(1 ) b→
(1 ) a→1
2
− − (1 − )
−∞
−∞
a
b
2
(1 ) 1
a
1 1
a ⎣ ⎦
b→ 2
(1 − ) b→ 1−
−∞
⎣ ⎦
2
dx dx dx
= lim + lim
x x x
∫ ∫ ∫
∫
dx ⎡ 1 ⎤ 1 1
= = −
−x ⎢ −x⎥
−b −a
b
dx ⎡ 1 ⎤
lim ∫ = lim − 0 =+∞ ( b 1)
x ⎢ b ⎥
<
dx ⎡ 1 ⎤
lim∫
= lim −1 − =+∞ (1 < a)
(1 −x) ⎢
⎣ 1−a⎥
⎦
a→1 2
a→1
a
b
(10)
− x −u
e dx e
2 −u
x = u , dx = 2 udu, = 2udu = 2e du
x u