integral from 0 to t of sin(a(t-u))

Lösung

\int_{0}^{t} sin (a (t - u)) d u

- \frac{- 1 + cos ( a t )}{a}

Schritte zur Lösung

\int_{0}^{t} sin (a (t - u)) d u

Wende U-Substitution an

= \int_{t}^{0} - sin (a v) d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{t}^{0} sin (a v) d v

Wende U-Substitution an

= - \int_{a t}^{0} \frac{sin ( w )}{a} d w

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{a} \cdot \int_{a t}^{0} sin (w) d w

Nutze das gemeinsame Integral :

\int sin (w) d w = - cos (w)

= - \frac{1}{a} [- cos (w)]_{a t}^{0}

Berechne die Grenzen:

- 1 + cos (a t)

= - \frac{1}{a} (- 1 + cos (a t))

Vereinfache

= - \frac{- 1 + cos ( a t )}{a}

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