Matematika 1B - sbírka - doc. Edita Kolářová (2010)
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u = sin x
v = x
v0 = 1
= x sin x −
Z
sin x dx =
= x sin x + cos x + C.
c)
Z
x
2 ln x dx =
u0 = x2
u =
x3
3
v = ln x v0 =
1
x
= . . . =
x3
3
ln x −
x3
9
+ C.
d)
Z
ln x dx =
u0 = 1
u = x
v = ln x v0 =
1
x
= x ln x −
Z
x
1
x
dx = x ln x − x + C.
e)
Z
arctg x dx =
u0 = 1
u = x
v = arctg x
v0 =
1
x2+1
= x arctg x−
Z
x
1
x2 + 1
dx =
= x arctg x −
1
2
Z
2x
x2 + 1
dx = x arctg x −
1
2
ln |x
2 + 1| + C.
f)
Z
ln(x
2+1) dx =
u0 = 1
u = x
v = ln(x2 + 1)
v0 =
2x
x2+1
= x ln(x
2+1)−
Z
x
2x
x2 + 1
dx =
= x ln(x
2 + 1) − 2
Z
x2
x2 + 1
dx = x ln(x
2 + 1) − 2
Z
x2 + 1 − 1
x2 + 1
dx =
= x ln(x
2 +1)−2
Z
1 dx+2
Z
1
x2 + 1
dx = x ln(x
2 + 1) − 2x + 2 arctg x + C.
g)
Z
(x
2 +1) ex dx =
u0 = ex
u = ex
v = x2 + 1
v0 = 2x
= (x
2 +1) ex −
Z
2x e
x dx =
=
u0 = ex
u = ex
v = 2x
v0 = 2
= (x
2 + 1) ex −
2x e
x −
Z
2 e
x dx
=
= (x
2 + 1) ex − 2x ex + 2 ex = ex (x2 − 2x + 3) + C.
h)
Z
(3x
2 − 4) sin x dx =
u0 = sin x
u = − cos x
v = 3x2 − 4
v0 =
6x
=
= −(3x
2 − 4) cos x +
Z
6x cos x dx =
u0 = cos x
u = sin x
v = 6x
v0 = 6
=
= −(3x
2 − 4) cos x + 6x sin x −
Z
6 sin x dx = 6x sin x + (10 − 3x
2) cos x + C.
j)
Z
(x
2 − 4x + 2) cos x dx =
u0 = cos x
u = sin x
v = x2 − 4x + 2
v0 =
2x − 4
=
= (x
2 −4x+2) sin x−
Z
(2x−4) sin x dx =
u0 = sin x
u = − cos x
v = 2x − 4
v0 =
2
=
(x
2−4x+2) sin x+(2x−4) cos x−
Z
2 cos x dx = (x