integral from 0 to 1 of xsin(nx)

解答

\int_{0}^{1} x sin (n x) d x

\frac{sin ( n )}{n ^{2}} - \frac{cos ( n )}{n}

求解步骤

\int_{0}^{1} x sin (n x) d x

使用分布积分法

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

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

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

化简

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

带入上下限计算后得:

\frac{1}{n ^{2}} sin (n) - \frac{1}{n} cos (n)

= \frac{1}{n ^{2}} sin (n) - \frac{1}{n} cos (n)

化简

= \frac{sin ( n )}{n ^{2}} - \frac{cos ( n )}{n}

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