Man berechne den Effektivwert Ieff{\displaystyle I_{e}ff} der Stromstärke eines Wechselstroms i(t)=i0sin(t){\displaystyle i(t)=i_{0}\sin(t)} gemäß
∫sin2xdx=12(x−sinx⋅cosx){\displaystyle \int \sin ^{2}x\,\mathrm {d} x={\frac {1}{2}}\left(x-\sin x\cdot \cos x\right)}
Ieff=12π∫02πi2(t)dt=12π∫02πi02⋅sin2(t)dt=i022π∫02πsin2(t)dt=i022π[12(t−sint⋅cost)]02π=i022π⋅2π2=i022=i02{\displaystyle {\begin{aligned}I_{eff}&={\sqrt {{\frac {1}{2\pi }}\int _{0}^{2\pi }i^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {1}{2\pi }}\int _{0}^{2\pi }i_{0}^{2}\cdot \sin ^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\int _{0}^{2\pi }\sin ^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\left[{\frac {1}{2}}\left(t-\sin t\cdot \cos t\right)\right]_{0}^{2\pi }}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\cdot {\frac {2\pi }{2}}}}\\&={\sqrt {\frac {i_{0}^{2}}{2}}}\\&={\frac {i_{0}}{\sqrt {2}}}\end{aligned}}}
![{\displaystyle {\begin{aligned}I_{eff}&={\sqrt {{\frac {1}{2\pi }}\int _{0}^{2\pi }i^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {1}{2\pi }}\int _{0}^{2\pi }i_{0}^{2}\cdot \sin ^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\int _{0}^{2\pi }\sin ^{2}(t)\,\mathrm {d} t}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\left[{\frac {1}{2}}\left(t-\sin t\cdot \cos t\right)\right]_{0}^{2\pi }}}\\&={\sqrt {{\frac {i_{0}^{2}}{2\pi }}\cdot {\frac {2\pi }{2}}}}\\&={\sqrt {\frac {i_{0}^{2}}{2}}}\\&={\frac {i_{0}}{\sqrt {2}}}\end{aligned}}}](/index.php?title=Spezial:MathShowImage&hash=e0bd91bc5a291f01910cec93892a1c3e&mode=mathml)