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

解答

\int_{0}^{1} x^{2} sin (π x) d x

\frac{1}{π} - \frac{4}{π ^{3}}

十进制

0.18930 \dots

求解步骤

\int_{0}^{1} x^{2} sin (π x) d x

使用分布积分法

= [- \frac{1}{π} x^{2} cos (π x) - \int - \frac{2}{π} x cos (π x) d x]_{0}^{1}

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

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

化简

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

带入上下限计算后得:

\frac{1}{π} - \frac{4}{π ^{3}}

= \frac{1}{π} - \frac{4}{π ^{3}}

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