Man berechne:
∫x⋅(lnx)2⋅dx{\displaystyle \int x\cdot (\ln {x})^{2}\cdot dx}
∫udv=uv−∫vdu{\displaystyle \int u\;dv=uv-\int v\;du} alias ∫u(x)v′(x)=u(x)v(x)−∫u′(x)v(x){\displaystyle \int u(x)\;v'(x)=u(x)v(x)-\int u'(x)v(x)}
Einfach ein par mal die Partielle Integration.
=12x2⋅(lnx)2−∫12⋅x2⋅2lnxx{\displaystyle ={\frac {1}{2}}x^{2}\cdot (\ln {x})^{2}-\int {\frac {1}{2}}\cdot x^{2}\cdot 2{\frac {\ln {x}}{x}}}
=12x2⋅(lnx)2−∫xlnx{\displaystyle ={\frac {1}{2}}x^{2}\cdot (\ln {x})^{2}-\int x\ln {x}}
=12x2⋅(lnx)2−12x2⋅lnx−∫12⋅x{\displaystyle ={\frac {1}{2}}x^{2}\cdot (\ln {x})^{2}-{\frac {1}{2}}x^{2}\cdot \ln {x}-\int {\frac {1}{2}}\cdot x}
=12x2⋅(lnx)2−12x2⋅lnx−14⋅x2{\displaystyle ={\frac {1}{2}}x^{2}\cdot (\ln {x})^{2}-{\frac {1}{2}}x^{2}\cdot \ln {x}-{\frac {1}{4}}\cdot x^{2}}
alias
