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

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

Ebaluatu

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

Ebazpidea

Ordeztu 2 * \pi balioa \tau balioarekin.

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

Ebaluatu lehenik integral indefinitua.

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

Aurkitu \frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}}en integrala integral arrunten taulako \int a\mathrm{d}\tau =a\tau araua erabiliz.

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

Sinplifikatu.

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

Hau da integral definitua: integrazioaren goiko limitean ebaluatutako adierazpenaren jatorrizko funtzioa ken integrazioaren beheko limitean ebaluatutako jatorrizko funtzioa.

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

Sinplifikatu.

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

Diferentziatu ξ balioarekiko

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