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Beliebt Rechnen >

integral from 0 to pi/2 of cos(5t)cos(10t)

Lösung

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t

Lösung

\frac{1}{15}

+1

Dezimale

0.06666 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{\frac{π}{2}} \frac{1}{2} (cos (15 t) + cos (- 5 t)) d t

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} cos (15 t) + cos (- 5 t) d t

Vereinfache

cos (15 t) + cos (- 5 t) : cos (5 t) + cos (15 t)

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} cos (5 t) + cos (15 t) d t

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{\frac{π}{2}} cos (5 t) d t + \int_{0}^{\frac{π}{2}} cos (15 t) d t)

\int_{0}^{\frac{π}{2}} cos (5 t) d t = \frac{1}{5}

\int_{0}^{\frac{π}{2}} cos (15 t) d t = - \frac{1}{15}

= \frac{1}{2} (\frac{1}{5} - \frac{1}{15})

Vereinfache

\frac{1}{2} (\frac{1}{5} - \frac{1}{15}) : \frac{1}{15}

= \frac{1}{15}

Graph

Beliebte Beispiele

(d^2)/(dx^2)(sin(3x^2+2))

\frac{d ^{2}}{d x ^{2}} (sin (3 x^{2} + 2))

limit as x approaches 0 of cos(x)-1

x \to 0 lim (cos (x) - 1)

derivative (sqrt(3))/(x^5)

d er i v a t i v e \frac{3}{x ^{5}}

integral of x^3sqrt(2+x^2)

\int x^{3} 2 + x^{2} d x

fläche y=2x-2x^2,[0,1]

a re a y = 2 x - 2 x^{2}, [0, 1]