Integration

$f\left(x\right)=x+1f\backslash left(x\backslash right)=x+1$

Integriere nun die Funktion.

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

$cc$ bestimmen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze den gegeben Punkt $\left(0\mid 0\right)\backslash left(\backslash left.0\backslash right|0\backslash right)$ in $F\left(x\right)F\backslash left(x\backslash right)$ ein.

$0=\frac{1}{2}{0}^2+0+c0=\backslash frac120^2+0+c$

$0=c0=c$

Stammfunktion aufstellen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze das gefundene $cc$ ein.

$F\left(x\right)=\frac{1}{2}{x}^2+x+0F\backslash left(x\backslash right)=\backslash frac12x^2+x+0$

Integration

$f\left(x\right)=x+1f\backslash left(x\backslash right)=x+1$

Integriere die Funktion.

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

c bestimmen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze den gegeben Punkt $\left(0\mathrm{\mid }1\right)\backslash left(0\backslash vert1\backslash right)$ in $F\left(x\right)F\backslash left(x\backslash right)$ ein.

$1=\frac{1}{2}{0}^2+0+c1=\backslash frac120^2+0+c$

$1=c1=c$

Stammfunktion aufstellen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze das gefundene $c=1c=1$ ein.

$F\left(x\right)=\frac{1}{2}{x}^2+x+1F\backslash left(x\backslash right)=\backslash frac12x^2+x+1$

Integration

$f\left(x\right)=x+1f\backslash left(x\backslash right)=x+1$

Integriere die Funktion.

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

$cc$ bestimmen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze den gegeben Punkt $\left(1\mathrm{\mid }0\right)\backslash left(1\backslash vert0\backslash right)$ in $F\left(x\right)F\backslash left(x\backslash right)$ ein.

$0=\frac{1}{2}{1}^2+1+c0=\backslash frac121^2+1+c$

$0=\frac{3}{2}+c0=\backslash frac32+c$

$c=-\frac{3}{2}c=-\backslash frac32$

Stammfunktion aufstellen

$F\left(x\right)=\frac{1}{2}{x}^2+x+cF\backslash left(x\backslash right)=\backslash frac12x^2+x+c$

Setze das gefundene $c=-\frac{3}{2}c=-\backslash frac\{3\}\{2\}$ ein.

$F\left(x\right)=\frac{1}{2}{x}^2+x-\frac{3}{2}F\backslash left(x\backslash right)=\backslash frac12x^2+x-\backslash frac32$