integral from 0 to a of xsin^2(pi/a x)

Lösung

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

\frac{a ^{2}}{4}

Schritte zur Lösung

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

\frac{π}{a} = π \frac{1}{a}

= \int_{0}^{a} x sin^{2} (π \frac{1}{a} x) d x

Multipliziere

π \frac{1}{a} x : \frac{π x}{a}

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

Wende die partielle Integration an

= [x (\frac{x}{2} - \frac{a}{4 π} sin (\frac{2 π x}{a})) - \int \frac{x}{2} - \frac{a}{4 π} sin (\frac{2 π x}{a}) d x]_{0}^{a}

\int \frac{x}{2} - \frac{a}{4 π} sin (\frac{2 π x}{a}) d x = \frac{x ^{2}}{4} + a \frac{a}{8 π ^{2}} cos (\frac{2 π x}{a})

= [x (\frac{x}{2} - \frac{a}{4 π} sin (\frac{2 π x}{a})) - (\frac{x ^{2}}{4} + a \frac{a}{8 π ^{2}} cos (\frac{2 π x}{a}))]_{0}^{a}

Vereinfache

[x (\frac{x}{2} - \frac{a}{4 π} sin (\frac{2 π x}{a})) - (\frac{x ^{2}}{4} + a \frac{a}{8 π ^{2}} cos (\frac{2 π x}{a}))]_{0}^{a} : [\frac{1}{8 π ^{2}} (2 π^{2} x^{2} - 2 πa x sin (\frac{2 π x}{a}) - a^{2} cos (\frac{2 π x}{a}))]_{0}^{a}

= [\frac{1}{8 π ^{2}} (2 π^{2} x^{2} - 2 πa x sin (\frac{2 π x}{a}) - a^{2} cos (\frac{2 π x}{a}))]_{0}^{a}

Berechne die Grenzen:

\frac{2 π ^{2} - 1}{8 π ^{2}} a^{2} + \frac{a ^{2}}{8 π ^{2}}

= \frac{2 π ^{2} - 1}{8 π ^{2}} a^{2} + \frac{a ^{2}}{8 π ^{2}}

Vereinfache

= \frac{a ^{2}}{4}