B - Prírodovedecká fakulta - Univerzita Komenského

a.
b.
c.
d.
e.
Riešenie.
∫ ∫
2 2
sin x = t t sin x
cos x.sin xdx = = tdt = + c = + c
cos xdx = dt 2 2
arcsinh x = t
dx = = tdt = + c = + c
2 2
arcsinh x t arcsin x
dx
2
1+ x
= dt 2 2
2
∫ ∫
1+ x
12 12
11 cos x = t 11 t cos x
∫ ∫
Lekcia 10 – Neurčitý integrál
cos x.sin x dx = = − t dt =− + c = − + c
− sin xdx = dt 12 12
tan x = t
2 2
tan x t tan x
dx = tdt c c
2 dx = = + = +
cos x = dt 2 2
2
cos x
∫ ∫
arcsin x = t
6 7 7
arcsin x 6 t arcsin x
dx = dx = t dt = + c = + c
2
1−x = dt 7 7
2
1−x ∫ ∫
4
4
5
sinh x = t
3 t sinh x
f. ∫ sinh x . cosh xdx
= = t dt = + c = + c
cosh xdx = dt ∫ 4 4
g.
h.
111 arctan x = t
112 112
arctan x 111 t arctan x
dx = t dt c c
2 dx = = + = +
1+ x = dt
112 112
2
1+ x
∫ ∫
∫
=
sin
∫
3
x.
cos
3
xdx
=
∫
sin
3
x.
sin x = t
2
3 2
4 5
( 1 − cos x)
cos xdx
=
= ∫t( 1 − t ) dt = ∫(
t − t )
cos xdx = dt
5 6
4 5 t t
5 ⎛ 1 t ⎞
5 ⎛ 1 sin x ⎞
t dt −∫
t dt = − + c1
+ c2
= t ⎜ − ⎟ + c = sin x⎜
− ⎟ + c
5 6
⎝ 5 6 ⎠ ⎝ 5 6 ⎠
125
dt =