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Beliebt Rechnen >

integral from 0 to 1 of x^2sin(npix)

Lösung

\int_{0}^{1} x^{2} sin (nπ x) d x

Lösung

\frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn}

Schritte zur Lösung

\int_{0}^{1} x^{2} sin (nπ x) d x

Wende die partielle Integration an

= [\frac{1}{3} x^{3} sin (πn x) - \int \frac{πn}{3} x^{3} cos (πn x) d x]_{0}^{1}

\int \frac{πn}{3} x^{3} cos (πn x) d x = \frac{1}{3} x^{3} sin (πn x) + \frac{1}{πn} x^{2} cos (πn x) - \frac{2}{π ^{2} n ^{2}} x sin (πn x) - \frac{2}{π ^{3} n ^{3}} cos (πn x)

= [\frac{1}{3} x^{3} sin (πn x) - (\frac{1}{3} x^{3} sin (πn x) + \frac{1}{πn} x^{2} cos (πn x) - \frac{2}{π ^{2} n ^{2}} x sin (πn x) - \frac{2}{π ^{3} n ^{3}} cos (πn x))]_{0}^{1}

Multipliziere aus

\frac{1}{3} x^{3} sin (πn x) - (\frac{1}{3} x^{3} sin (πn x) + \frac{1}{πn} x^{2} cos (πn x) - \frac{2}{π ^{2} n ^{2}} x sin (πn x) - \frac{2}{π ^{3} n ^{3}} cos (πn x)) : - \frac{1}{πn} x^{2} cos (πn x) + \frac{2}{π ^{2} n ^{2}} x sin (πn x) + \frac{2}{π ^{3} n ^{3}} cos (πn x)

= [- \frac{1}{πn} x^{2} cos (πn x) + \frac{2}{π ^{2} n ^{2}} x sin (πn x) + \frac{2}{π ^{3} n ^{3}} cos (πn x)]_{0}^{1}

Berechne die Grenzen:

(- 1)^{n} \frac{2}{π ^{3} n ^{3}} - (- 1)^{n} \frac{1}{πn} - \frac{2}{π ^{3} n ^{3}}

= (- 1)^{n} \frac{2}{π ^{3} n ^{3}} - (- 1)^{n} \frac{1}{πn} - \frac{2}{π ^{3} n ^{3}}

Vereinfache

= \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn}

Beliebte Beispiele

integral of tan^2(3x-1)

\int tan^{2} (3 x - 1) d x

integral of 5x^{-6/7}

\int 5 x^{- \frac{6}{7}} d x

limit as x approaches 3 of (x-2)/(6-2x)

x \to 3 lim (\frac{x - 2}{6 - 2 x})

derivative y= 1/(cos(x/2))

d er i v a t i v e y = \frac{1}{cos ( \frac{x}{2} )}

integral of 216x^2cos(6x)

\int 216 x^{2} cos (6 x) d x