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Popolare Calcoli >

integral from 0 to l of xsin^2((pix)/l)

Soluzione

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

Soluzione

\frac{l ^{2}}{4}

Fasi della soluzione

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

Applica l'Integrazione per Parti

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

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

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

Semplificare

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

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

Calcola i limiti:

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

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

Semplificare

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

Esempi popolari

pendenza (5.1)(3.6)

s l o p e (5.1) (3.6)

integral of (7x+2)/(x^2-4)

\int \frac{7 x + 2}{x ^{2} - 4} d x

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\frac{d}{d x} (\frac{( e ^{x} - e ^{- x} )}{( e ^{x} + e ^{- x} )})

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\int e^{x} sin^{2} (x) d x

derivative (10x+4x^2)/((5+4x)^2)

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