integral from 0 to 1 of xcos(x^4)

解

\int_{0}^{1} x cos (x^{4}) d x

\frac{π C ( 1 \cdot \frac{2}{π} ) - π C ( 0 \cdot \frac{2}{π} )}{2 2}

解答ステップ

\int_{0}^{1} x cos (x^{4}) d x

部分積分を適用する

= [\frac{1}{2} x^{2} cos (x^{4}) - \int - 2 x^{5} sin (x^{4}) d x]_{0}^{1}

\int - 2 x^{5} sin (x^{4}) d x = - \frac{1}{2} (- x^{2} cos (x^{4}) + \frac{π}{2} C (\frac{2}{π} x^{4}))

= [\frac{1}{2} x^{2} cos (x^{4}) - (- \frac{1}{2} (- x^{2} cos (x^{4}) + \frac{π}{2} C (\frac{2}{π} x^{4})))]_{0}^{1}

簡素化

[\frac{1}{2} x^{2} cos (x^{4}) - (- \frac{1}{2} (- x^{2} cos (x^{4}) + \frac{π}{2} C (\frac{2}{π} x^{4})))]_{0}^{1} : [\frac{π}{2 2} C (\frac{2}{π} x^{4})]_{0}^{1}

= [\frac{π}{2 2} C (\frac{2}{π} x^{4})]_{0}^{1}

限度を計算する:

\frac{π C ( 1 \cdot \frac{2}{π} ) - π C ( 0 \cdot \frac{2}{π} )}{2 2}

= \frac{π C ( 1 \cdot \frac{2}{π} ) - π C ( 0 \cdot \frac{2}{π} )}{2 2}