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# Appendix E: Key Integrals

${V}_{average}=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}v\text{\hspace{0.17em}}d\omega t$

${V}_{rms}=\sqrt{\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{v}^{2}\text{\hspace{0.17em}}d\omega t}$

$\int {e}^{ax}\mathrm{cos}\left(bx\right)=\frac{{e}^{ax}}{{a}^{2}+{b}^{2}}\left[a\text{\hspace{0.17em}}\mathrm{cos}\left(bx\right)+b\text{\hspace{0.17em}}\mathrm{sin}\left(bx\right)\right]$

$\underset{t=0}{\overset{24}{\int }}{e}^{\frac{-{\left(t-{t}_{o}\right)}^{2}}{2{\sigma }^{2}}}dt\approx \sqrt{2\pi }\sigma$

$\int {e}^{ax}\left[c×\mathrm{cos}\left(bx\right)+k×\mathrm{sin}\left(bx\right)\right]dx=\frac{{e}^{ax}}{{a}^{2}+{b}^{2}}\left[\left(ka+cb\right)\mathrm{sin}\left(bx\right)+\left(ca-kb\right)\text{\hspace{0.17em}}\mathrm{cos}\left(bx\right)\right]$

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