Berechnen Sie die Stammfunktionen der folgenden Funktionen. A) b ( v ) = 7 v 5 {\displaystyle \ b(v)=7\ v^{5}\quad } B) a ( c ) = b c d {\displaystyle \ a(c)=b\ c^{d}} C) f ( x ) = 65 x 12 {\displaystyle \ f(x)=65\ x^{12}} D) x ( y ) = y 2 − 2 x + 1 {\displaystyle \ x(y)=y^{2}-2x+1\quad } E) v ( t ) = t 3 4 {\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\quad } F) f ( x ) = 4 x 5 {\displaystyle \ f(x)={\frac {4}{x^{5}}}\quad } G) V ( h ) = 1 h 5 4 {\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\quad } H) H ( x ) = 27 x 7 3 {\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\quad } I) z ( w ) = 5 w {\displaystyle \ z(w)={\frac {5}{w}}\ } J) m ( n ) = 5 n + 3125 n 30 5 − 5 13 n 12 {\displaystyle \ m(n)={\frac {5}{n}}+{\sqrt[{5}]{\frac {3125}{n^{30}}}}-{\frac {5}{13}}n^{12}\ } .
Regel: ∫ a x n d x = a x n + 1 n + 1 + c {\displaystyle \ \int ax^{n}dx={\frac {a\ x^{n+1}}{n+1}}\ +c\ \quad }
b ( v ) = 7 v 5 ⇒ ∫ b ( v ) d v = 7 6 v 6 + c {\displaystyle \ b(v)=7\ v^{5}\ \Rightarrow \ \int b(v)dv={\frac {7}{6}}\ v^{6}+c\ \quad } Hoch zum Anfang
a ( c ) = b c d ⇒ ∫ a ( c ) d c = b d + 1 c d + 1 + k {\displaystyle \ a(c)=b\ c^{d}\ \Rightarrow \ \int a(c)dc={\frac {b}{d+1}}\ c^{d+1}+k} Hoch zum Anfang
f ( x ) = 65 x 12 ⇒ ∫ f ( x ) d x = 65 ⋅ x 12 + 1 12 + 1 = 5 x 13 + c {\displaystyle \ f(x)={65}\ x^{12}\ \Rightarrow \ \int f(x)dx=65\cdot {\frac {x^{12+1}}{12+1}}=5\ x^{13}+c\ \quad } Hoch zum Anfang
x ( y ) = y 2 − 2 y + 1 ⇒ ∫ x ( y ) d y = y 3 3 − 2 ⋅ y 2 2 + y + c {\displaystyle \ x(y)=y^{2}-2y+1\ \Rightarrow \ \int x(y)dy={\frac {y^{3}}{3}}\ -2\cdot {\frac {y^{2}}{2}}\ +y+c\ \quad } Hoch zum Anfang
v ( t ) = t 3 4 = t 3 4 ⇒ ∫ v ( t ) d t = ( t 3 4 + 1 3 4 + 1 ) + c {\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\ =t^{\frac {3}{4}}\ \ \Rightarrow \ \int v(t)dt=\left({\dfrac {t^{{\frac {3}{4}}+1}}{{\frac {3}{4}}+1}}\right)+c\ }
= 4 7 t 7 4 + c ( = 4 t 7 4 7 + c ) {\displaystyle \qquad ={\frac {4}{7}}t^{\frac {7}{4}}+c\left(={\frac {4\ {\sqrt[{4}]{t^{7}}}}{7}}+c\right)\quad } Hoch zum Anfang
f ( x ) = 4 x 5 = 4 x − 5 ⇒ ∫ f ( x ) d x = {\displaystyle \ f(x)={\frac {4}{x^{5}}}=4\ x^{-5}\ \ \Rightarrow \ \int f(x)dx=}
4 ⋅ x − 5 + 1 − 5 + 1 ) = − x − 4 + c {\displaystyle \qquad {\frac {4\cdot x^{-5+1}}{-5+1}})\ =-x^{-4}+c\quad } Hoch zum Anfang
V ( h ) = 1 h 5 4 = h − 5 4 = h − 5 4 ⇒ {\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\ ={\sqrt[{4}]{h^{-5}}}\ =h^{-{5 \over 4}}\ \Rightarrow \ }
∫ V ( h ) d h = h − 5 4 + 1 − 5 4 + 1 + c = − 4 h − 1 4 + c ( = − 4 h 4 + c ) {\displaystyle \qquad \int V(h)dh={\dfrac {h^{-{\frac {5}{4}}+1}}{-{\frac {5}{4}}+1}}+c\ =-{4}\ h^{-{\frac {1}{4}}}+c\left(=-{\frac {4}{\sqrt[{4}]{h}}}+c\right)} Hoch zum Anfang
H ( x ) = 27 x 7 3 = 27 3 x − 7 3 = 3 x − 7 3 ⇒ {\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\ ={\sqrt[{3}]{27}}{\sqrt[{3}]{x^{-7}}}=3\ x^{-{\frac {7}{3}}}\ \Rightarrow \ }
∫ H ( x ) d x = 3 ⋅ x − 7 3 + 1 − 7 3 + 1 + c = − 9 4 x − 4 3 + c ( = − 9 4 x 4 3 + c ) {\displaystyle \qquad \int H(x)dx={\dfrac {3\cdot x^{-{\frac {7}{3}}+1}}{-{\frac {7}{3}}+1}}+c=-{\frac {9}{4}}\ x^{-{\frac {4}{3}}}+c\left(=-{\frac {9}{4\ {\sqrt[{3}]{x^{4}\ }}}}+c\right)} Hoch zum Anfang
z ( w ) = 5 w = 5 ⋅ w − 1 ⇒ ∫ z ( w ) d w = 5 ln w + c {\displaystyle \ z(w)={\frac {5}{w}}=5\cdot w^{-1}\ \Rightarrow \ \int z(w)dw=5\ln w+c\ \quad } Hoch zum Anfang
m ( n ) = 5 n − 3125 n 30 5 − 13 n 12 {\displaystyle \ m(n)={\frac {5}{n}}-{\sqrt[{5}]{\frac {3125}{n^{30}}}}-{13}\ n^{12}\ }
1 3125 n 30 5 = 1 3125 5 n − 30 5 = 1 5 n − 30 5 = 1 5 x − 6 {\displaystyle \qquad {\sqrt[{5}]{\frac {1}{3125\ n^{30}}}}\ ={\sqrt[{5}]{{\frac {1}{3125}}\ }}\ {\sqrt[{5}]{n^{-30}\ \ }}={\frac {1}{5}}\ n^{-{\frac {30}{5}}}={\frac {1}{5}}\ x^{-6}\ }
m ( n ) = 5 n − 5 x − 6 − 13 n 12 {\displaystyle \qquad m(n)={\frac {5}{n}}-5\ x^{-6}-{13}\ n^{12}\ }
∫ m ( n ) d n = 5 ln n − 5 ⋅ n − 6 + 1 − 6 + 1 − 13 ⋅ n 12 + 1 12 + 1 + c {\displaystyle \qquad \int m(n)dn=5\ \ln n-5\cdot {\frac {n^{-6+1}}{-6+1}}\ -13\cdot {\frac {n^{12+1}}{12+1}}\ +c}
∫ m ( n ) d n = 5 ln n + n 5 − n 13 + c {\displaystyle \qquad \int m(n)dn=5\ln n+n^{5}-n^{13}+c} Hoch zum Anfang
![{\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\ =t^{\frac {3}{4}}\ \ \Rightarrow \ \int v(t)dt=\left({\dfrac {t^{{\frac {3}{4}}+1}}{{\frac {3}{4}}+1}}\right)+c\ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/f7d1df42f27df8ef3e819117557ed86f1d714fba.svg)
![{\displaystyle \qquad ={\frac {4}{7}}t^{\frac {7}{4}}+c\left(={\frac {4\ {\sqrt[{4}]{t^{7}}}}{7}}+c\right)\quad }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/10c06c7cb65071fc45c16e81aa14b8ac4e33f25b.svg)
![{\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\ ={\sqrt[{4}]{h^{-5}}}\ =h^{-{5 \over 4}}\ \Rightarrow \ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/4e3c2e9e70d29dd41eb1eb9b9cde7a4b81b20bc4.svg)
![{\displaystyle \qquad \int V(h)dh={\dfrac {h^{-{\frac {5}{4}}+1}}{-{\frac {5}{4}}+1}}+c\ =-{4}\ h^{-{\frac {1}{4}}}+c\left(=-{\frac {4}{\sqrt[{4}]{h}}}+c\right)}](../_assets_/eb734a37dd21ce173a46342d1cc64c92/13f1100886f24da3036922ee6cf1334841c93583.svg)
![{\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\ ={\sqrt[{3}]{27}}{\sqrt[{3}]{x^{-7}}}=3\ x^{-{\frac {7}{3}}}\ \Rightarrow \ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/2fc4cabc121b43763b96ee2fe5b9d05f641124da.svg)
![{\displaystyle \qquad \int H(x)dx={\dfrac {3\cdot x^{-{\frac {7}{3}}+1}}{-{\frac {7}{3}}+1}}+c=-{\frac {9}{4}}\ x^{-{\frac {4}{3}}}+c\left(=-{\frac {9}{4\ {\sqrt[{3}]{x^{4}\ }}}}+c\right)}](../_assets_/eb734a37dd21ce173a46342d1cc64c92/8a2cd6f0a21457bd0fcc5c66f5d8d96b412f0da4.svg)
![{\displaystyle \ m(n)={\frac {5}{n}}-{\sqrt[{5}]{\frac {3125}{n^{30}}}}-{13}\ n^{12}\ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/a8dc341099c2c41f3671dc7806431715eefffd9e.svg)
![{\displaystyle \qquad {\sqrt[{5}]{\frac {1}{3125\ n^{30}}}}\ ={\sqrt[{5}]{{\frac {1}{3125}}\ }}\ {\sqrt[{5}]{n^{-30}\ \ }}={\frac {1}{5}}\ n^{-{\frac {30}{5}}}={\frac {1}{5}}\ x^{-6}\ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/a460b2c6207bb5a0f62a233254df658cbeabe427.svg)
![{\displaystyle \ v(t)={\sqrt[{4}]{t^{3}}}\quad }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/d526346eb3a2a98cbf0efb74449697145f89a41f.svg)
![{\displaystyle \ V(h)={\sqrt[{4}]{\frac {1}{h^{5}}}}\quad }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/adbec42ff2d23596366bdd416a557e93d4761e91.svg)
![{\displaystyle \ H(x)={\sqrt[{3}]{\frac {27}{x^{7}}}}\quad }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/daa4a072dbb975f068d96688396d3124b06c53b3.svg)
![{\displaystyle \ m(n)={\frac {5}{n}}+{\sqrt[{5}]{\frac {3125}{n^{30}}}}-{\frac {5}{13}}n^{12}\ }](../_assets_/eb734a37dd21ce173a46342d1cc64c92/1e756831d09dfcab88c4a3d73fe7132769ff362b.svg)