1 Neurčitý integrál
1.1 Základní vzorce (platí na definičním oboru integrované funkce)
c dx = cx + C; k ∈ R speciálně 1 dx = x + C (1)
xn
dx =
xn+1
n + 1
+ C pro x > 0, n ∈ R, n = −1 (2)
1
x
dx = ln |x| + C pro x = 0 (3)
ex
dx = ex
+ C (4)
ax
dx =
ax
ln a
+ C pro a > 0, a = 1 (5)
sin x dx = − cos x + C (6)
cos x dx = sin x + C (7)
1
cos2 x
dx = tg x + C pro x =
π
2
+ kπ, k ∈ Z (8)
1
sin2
x
dx = − cotg x + Cpro x = kπ, k ∈ Z (9)
f (x)
f(x)
dx = ln |f(x)| + C (10)
1.2 Operace s integrály
c · f(x) dx = c · f(x) dx = c · F(x) + C (11)
(f(x) ± g(x)) dx = f(x) dx ± g(x) dx = F(x) ± G(x) + C (12)
1.3 Per partes
u(x) · v (x) dx = u(x) · v(x) − u (x) · v(x) dx (13)
u (x) · v(x) dx = u(x) · v(x) − u(x) · v (x) dx (14)
1.4 Substituce
f(g(x))
f(t)
· g (x) dx
dt
=
t = g(x)
dt = g (x) dx
= f(t) dt = F(g(x))
F (t)
+C (15)
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