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Beliebt Rechnen >

integral from 0 to a/2 of sin^2((pix)/a)

Lösung

\int_{0}^{\frac{a}{2}} sin^{2} (\frac{π x}{a}) d x

Lösung

\frac{a}{4}

Schritte zur Lösung

\int_{0}^{\frac{a}{2}} sin^{2} (\frac{π x}{a}) d x

Wende U-Substitution an

= \int_{0}^{\frac{π}{2}} \frac{a sin ^{2} ( u )}{π} d u

Entferne die Konstante:

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

= \frac{a}{π} \cdot \int_{0}^{\frac{π}{2}} sin^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{a}{π} \cdot \int_{0}^{\frac{π}{2}} \frac{1 - cos ( 2 u )}{2} d u

Entferne die Konstante:

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

= \frac{a}{π} \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (2 u) d u

Wende die Summenregel an:

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

= \frac{a}{π} \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d u - \int_{0}^{\frac{π}{2}} cos (2 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (2 u) d u = 0

= \frac{a}{π} \cdot \frac{1}{2} (\frac{π}{2} - 0)

Vereinfache

\frac{a}{π} \cdot \frac{1}{2} (\frac{π}{2} - 0) : \frac{a}{4}

= \frac{a}{4}

Beliebte Beispiele

integral from-2 to 1 of 3+y^2

\int_{- 2}^{1} 3 + y^{2} d y

integral from pi/6 to pi/4 of sin(4t)

\int_{\frac{π}{6}}^{\frac{π}{4}} sin (4 t) d t

integral from 0 to 8 of 9e^x

\int_{0}^{8} 9 e^{x} d x

integral from-5 to-2 of-x-3

\int_{- 5}^{- 2} - x - 3 d x

integral from 3 to 6 of x^2-x+1

\int_{3}^{6} x^{2} - x + 1 d x