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

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

I-evaluate

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

Mga Hakbang ng Solution

I-substitute ang 2 * \pi para sa \tau.

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

Suriin muna ang indefinite na integral.

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

Hanapin ang integral ng \frac{\sin(\omega _{d}\left(t-2\pi \right))}{e^{\xi \omega \left(t-2\pi \right)}} gamit ang alituntunin ng talahanayan ng mga karaniwang integral \int a\mathrm{d}\tau =a\tau .

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

Pasimplehin.

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

Ang definite integral ay ang antiderivative ng expression na na-evaluate sa upper limit ng integration minus ang antiderivative na na-evaluate sa lower limit ng integration.

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

Pasimplehin.

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

I-differentiate ang w.r.t. ξ

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