integral de 0 a 1 de x^2sin(npix)

Solução

\int_{0}^{1} x^{2} sin (nπ x) d x

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

Passos da solução

\int_{0}^{1} x^{2} sin (nπ x) d x

Aplicar integração por partes

= [\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}

Expandir

\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}

Calcular os limites:

(- 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}}

Simplificar

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

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