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p.56 – 36a
∫ x ⋅ ln ( x ) dx
rule: ∫ f ( x ) ⋅ g' ( x ) dx = f ( x ) ⋅ g ( x ) − ∫ f ' ( x ) ⋅ g ( x ) dx
1
x
f ( x ) = ln ( x ) ⇒ f ' ( x ) =
1 2
x
2
1 2
1 1 2
∫ ln ( x ) ⋅ x dx = ln ( x ) ⋅ 2 x − ∫ x ⋅ 2 x dx
1
1
= x 2 ⋅ ln ( x ) − ∫ x dx
2
2
1 2
1
= x ⋅ ln ( x ) − x 2 + c
2
4
1 2 ⎛
1⎞
= x ⋅ ⎜ ln ( x ) − ⎟ + c
⎝
2
2⎠
g' ( x ) = x ⇒ g ( x ) =
p.56 – 36b
∫ x ⋅ cos ( x ) dx
rule: ∫ f ( x ) ⋅ g' ( x ) dx = f ( x ) ⋅ g ( x ) − ∫ f ' ( x ) ⋅ g ( x ) dx
f ( x ) = x ⇒ f '( x ) = 1
g' ( x ) = cos ( x ) ⇒ g ( x ) = sin ( x )
∫ x ⋅ cos ( x ) dx = x ⋅sin ( x ) − ∫ 1⋅sin ( x ) dx
= x ⋅sin ( x ) + cos ( x ) + c
p.56 – 36c
∫ x⋅e
rule:
x
dx
∫ f ( x ) g'( x ) dx = f ( x ) g ( x ) − ∫ f '( x ) g ( x ) dx
f ( x ) = x ⇒ f '( x ) = 1
g' ( x ) = e x ⇒ g ( x ) = e x
∫ x⋅e
x
dx = x ⋅ e x − ∫ e x dx
= x ⋅ ex − ex + c
= e x ⋅ ( x − 1) + c
p.56 – 36e
∫ ln ( x ) dx = ∫ 1⋅ ln ( x ) dx
rule: ∫ f ( x ) g' ( x ) dx = f ( x ) g ( x ) − ∫ f ' ( x ) g ( x ) dx
f ( x ) = ln ( x ) ⇒ f ' ( x ) =
g' ( x ) = 1 ⇒ g ( x ) = x
1
x
1
∫ ln ( x ) dx = x ⋅ ln ( x ) − ∫ x ⋅ x dx
= x ⋅ ln ( x ) − x + c
x ⋅ ( ln ( x ) − 1) + c
1/2
p.56 – 37a
∫x
2
⋅ e x dx
rule:
∫ f ( x ) g'( x ) dx = f ( x ) g ( x ) − ∫ f '( x ) g ( x ) dx
f ( x ) = x 2 ⇒ f ' ( x ) = 2x
g' ( x ) = e x ⇒ g ( x ) = e x
∫x
2
⋅ e x dx = x 2 ⋅ e x − ∫ 2x ⋅ e x dx
= x 2 ⋅ e x − 2 ⋅ ∫ x ⋅ e x dx
= x 2 ⋅ e x − 2 ⋅ ( x − 1) ⋅ e x + c
(
)
= x 2 − 2x + 2 ⋅ e x + c
p.56 – 37b
∫ sin ( x ) ⋅ e dx
rule: ∫ f ( x ) g' ( x ) dx = f ( x ) g ( x ) − ∫ f ' ( x ) g ( x ) dx
x
f ( x ) = sin ( x ) ⇒ f ' ( x ) = cos ( x )
g' ( x ) = e x ⇒ g ( x ) = e x
∫ sin ( x ) ⋅ e
x
dx = sin ( x ) ⋅ e x − ∫ cos ( x ) ⋅ e x dx
!##"##
$
(1)
(1):
u ( x ) = cos ( x ) ⇒ u ' ( x ) = − sin ( x )
v' ( x ) = e x ⇒ v ( x ) = e x
∫ cos ( x ) ⋅ e
x
dx = cos ( x ) ⋅ e x − ∫ − sin ( x ) ⋅ e x dx
∫ sin ( x ) ⋅ e
∫ sin ( x ) ⋅ e
∫ sin ( x ) ⋅ e
2 ⋅ ∫ sin ( x ) ⋅ e
∫ sin ( x ) ⋅ e
= cos ( x ) ⋅ e x + ∫ sin ( x ) ⋅ e x dx
x
x
x
dx = sin ( x ) ⋅ e x − ∫ cos ( x ) ⋅ e x dx
(
dx = sin ( x ) ⋅ e x − cos ( x ) ⋅ e x + ∫ sin ( x ) ⋅ e x dx
)
dx = sin ( x ) ⋅ e x − cos ( x ) ⋅ e x − ∫ sin ( x ) ⋅ e x dx
x
dx = sin ( x ) ⋅ e x − cos ( x ) ⋅ e x + c
x
dx =
1 x
⋅ e ⋅ ( sin ( x ) − cos ( x )) + c
2
2/2