Integration durch lineare Substitution

f(x)\; =\; \;  \sqrt{ x } 

linke Integrationsgrenze für h(x): a = 

rechte Integrationsgrenze für h(x): b = 

Innere Funktion g(x)= mx + c :

Substitution durchführen

anklicken!

(haj v 1.0)

b = 8 (slider from -3 to 20, step 0.25)
f(x) = sqrt(x)
m = 1 (slider from -2 to 2, step 0.25)
c = -4.5 (slider from -12 to 6, step 0.5)
gb: m b + c
a = 5 (slider from -3 to 20, step 0.25)
I_{subs} = Integral(f, m a + c, m b + c)
Wert: I_{subs} / m
ga: m a + c
iw = Integral(f, ga, gb)
g(x) = m x + c
h(x) = f(g(x))
I_{ab} = Integral(h, a, b)
T2: "Bsp.:  \;  \;  \; h(x) \;  \;= \;  \;  \sqrt{ mx \; + \; c  }\; =\; \;   \sqrt{ " + m + "\;x +\; (" + c + ") }"
lks = Segment((a, 0), (a, h(a)))
rts = Segment((b, 0), (b, h(b)))
T7: "m a + c = " + m + " a + (" + c + ") = " + (m a) + " + (" + c + ") = " + ga
T8: "m b + c = " + m + " b + (" + c + ") = " + (m b) + " + (" + c + ") = " + gb
T9: "Gesucht: \; \; \; \int_{a }^{b }h(x)dx \;= \; \; \int_{" + a + " }^{" + b + " } \; \sqrt{ mx \;+ \;c  }\;dx \;= \; \;\int_{" + a + " }^{" + b + " } \; \sqrt{ " + m + "x \;+ \;(" + c + ")  }\;dx"
T10: " Substitution: \; \; \; 1/m\; *\int_{ma\; +c }^{mb\; +c }f(x)dx\; =  1/m *\int_{" + ga + " }^{" + gb + " } \; \sqrt{ x  }\; dx\; =\; \; 1/" + m + "\;*\;" + iw + "\;=\;\;" + Wert
d = false
s1 = Segment((a, h(a)), (ga, f(ga)))
s2 = Segment((b, h(b)), (gb, f(gb)))