1.4.2. Osittaisintegroint
f ´( x )· g ( x ) dx f ( x ) g ( x ) f ( x ) g ' ( x ) dx
Perustelu:
)) ( ) (
( f x g x
D f ' ( x ) g ( x ) f ( x ) g ' ( x )
Integroidaan puolittain:
) ( )
( x g x
f f ' ( x ) g ( x ) dx f ( x ) g ' ( x ) dx
f ' ( x ) g ( x ) dx f ( x ) g ( x ) f ( x ) g ' ( x ) dx
f´(x)·g(x)dx f(x)g(x) f (x)g'(x)dx E.3. Laske a) E.3. ex·2xdx b)
xe2xdxf ’ (x) = ex f (x) = ex + C C = 0:
f(x) = ex
g (x) = 2x g ’(x) = 2
e
x· 2 xdx e
x 2 x e
x 2 dx e
x 2 x 2 e
xdx
C e
x
e
x
x
2 2
C e
x
x
2 ( 1 )
f´(x)·g(x)dx f(x)g(x) f (x)g'(x)dx
xe2xdxf ’ (x) = e2x f (x) = ½e2x
g (x) = x g ’(x) = 1
xe
2xdx ½ e
2x x ½ e
2x 1 dx 1 2 e
2x x 4 1 2 e
2xdx
C e
x
e
x
x
2 24 1 2
1
b)
C e
xe
x
x
2 24 1 2
1
( 3 x 4 ) e
xdx
E.4.E.4.
f´(x)·g(x)dx f(x)g(x) f (x)g'(x)dx f ’ (x) = ex
f (x) = ex
g (x) = 3x + 4 g ’(x) = 3
( 3 x 4 ) e
xdx e
x ( 3 x 4 ) e
x 3 dx e
x ( 3 x 4 ) 3 e
xdx
C e
x
e
x
x
( 3 4 ) 3 C e
x
x
( 3 1 )
E.5.E.5.
x 2x 1dx f´(x)·g(x)dx f (x)g(x) f (x)g'(x)dxf ’ (x) = f (x) =
g (x) = x g ’(x) = 1
x 2 x 1 dx 1 3 ( 2 x 1 )
3/2 x 1 3 ( 2 x 1 )
3/2 1 dx
2 ( 2 x 1 )
3/2dx
6 1
)½
1 2
( 1
2x x
2 /
)3
1 2
3(
1 x
( 2 x 1 )
3/2x 3
1
( 2 x 1 )
3/2x 3
1
25
) 1 2
( 2 1 3
1 6
1
x
) 1 3
( 1 2
) 1 2
15 (
1
x x x
( 2 x 1 )
3/2x 3
1
25