integral from 1/2 to 1 of xcos(pix)

解答

\int_{\frac{1}{2}}^{1} x cos (π x) d x

- \frac{1}{π ^{2}} - \frac{1}{2 π}

十进制

- 0.26047 \dots

求解步骤

\int_{\frac{1}{2}}^{1} x cos (π x) d x

使用分布积分法

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

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

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

化简

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

带入上下限计算后得:

- \frac{1}{π ^{2}} - \frac{1}{2 π}

= - \frac{1}{π ^{2}} - \frac{1}{2 π}

\int_{0}^{5} \frac{1}{9 + 4 x ^{2}} d x

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