A4-format til udskrift. - Aarhus Universitet

82 II. INTEGRATION
Bevis
g ′ g(x + h) − g(x)
(x) = lim
h→0
h
x+h
1
= lim f(u)du
h→0 h x
f(x
= lim
h→0
∗ )h
h
= f(x)
2.4. Stamfunktions botanik ☞ [S] 5.3 Evaluating definite integrals. . .
1 Stamfunktioner
x n dx = 1
n + 1 xn+1 , n = −1
1
dx = ln(x)
x
e x dx = e x
ln(x)dx = xln(x) − x
a x dx = 1
ln(a) ax
2.5. Stamfunktions botanik ☞ [S] 5.3 Evaluating definite integrals. . .
1 Stamfunktioner - flere
sin(x)dx = −cos(x)
cos(x)dx = sin(x)
1
√ 1 − x 2 dx = sin−1 (x)
1
1 + x 2 dx = tan−1 (x)
2.6. Integral smertefrit ☞ [S] 5.4 The fundamental theorem of calculus
Eksempel
Beregn
4
(x
1
2 + x 3 )dx