Rešite ∫ (from 0 to 1) of e^-xiomega(t-tau)sin(omega_{d}(t-tau))dtau

$$\int _ { 0 } ^ { 1 } e ^ { - \xi \omega ( t - \tau ) } \sin ( \omega _ { d } ( t - \tau ) ) d \tau$$

Ovrednoti

$$\frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}$$

Koraki rešitve

Nadomestek 2 * \pi za \tau.

$$\int_{0}^{1} {e ^ {-\xi \omega {(t - 2 * \pi)}} \sin(\omega_{d} {(t - 2 * \pi)})} d\tau$$

Najprej ovrednotite nedoločni integral.

$$\int \frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}\mathrm{d}\tau$$

Poiščite integral \frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}} s tabelo skupnega integrali pravila \int a\mathrm{d}\tau =a\tau .

$$\frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}\tau$$

Poenostavite.

$$\frac{\sin(\omega _{d}\left(t-2\pi \right))\tau }{e^{\xi \omega \left(t-2\pi \right)}}$$

Določen integral je integral izraza, ovrednotenega pri zgornji omejitvi integriranja, minus integral, ovrednoten pri spodnji omejitvi integriranja.

$$e^{-\xi \omega \left(t-2\pi \right)}\sin(\omega _{d}\left(t-2\pi \right))+0e^{-\xi \omega \left(t-2\pi \right)}\sin(\omega _{d}\left(t-2\pi \right))$$

Poenostavite.

$$\frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}$$

Odvajajte w.r.t. ξ

$$-\frac{\omega \left(t-2\pi \right)\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}$$