integral from 0 to pi of sin(x)x^5

解

\int_{0}^{π} sin (x) x^{5} d x

π^{5} - 20 (π^{3} - 6 π)

十進法表記

62.88526 \dots

解答ステップ

\int_{0}^{π} sin (x) x^{5} d x

部分積分を適用する

= [- x^{5} cos (x) - \int - 5 x^{4} cos (x) d x]_{0}^{π}

\int - 5 x^{4} cos (x) d x = - 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))))

= [- x^{5} cos (x) - (- 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))))]_{0}^{π}

簡素化

= [- x^{5} cos (x) + 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))))]_{0}^{π}

限度を計算する:

π^{5} - 20 (π^{3} - 6 π)

= π^{5} - 20 (π^{3} - 6 π)