integral from 0 to pi of x^3sin(nx)

解答

\int_{0}^{π} x^{3} sin (n x) d x

\frac{6 π ( - 1 ) ^{n} - π ^{3} ( - 1 ) ^{n} n ^{2}}{n ^{3}}

求解步骤

\int_{0}^{π} x^{3} sin (n x) d x

使用分布积分法

= [\frac{1}{n} (- x^{3} cos (n x) - n \cdot \int - \frac{3}{n} x^{2} cos (n x) d x)]_{0}^{π}

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

= [\frac{1}{n} (- x^{3} cos (n x) - n (- \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x)))))]_{0}^{π}

化简

[\frac{1}{n} (- x^{3} cos (n x) - n (- \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x)))))]_{0}^{π} : [\frac{1}{n} (- x^{3} cos (n x) + \frac{3}{n ^{3}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x))))]_{0}^{π}

= [\frac{1}{n} (- x^{3} cos (n x) + \frac{3}{n ^{3}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x))))]_{0}^{π}

带入上下限计算后得:

\frac{1}{n} (\frac{6 π ( - 1 ) ^{n}}{n ^{2}} - π^{3} (- 1)^{n})

= \frac{1}{n} (\frac{6 π ( - 1 ) ^{n}}{n ^{2}} - π^{3} (- 1)^{n})

化简

= \frac{6 π ( - 1 ) ^{n} - π ^{3} ( - 1 ) ^{n} n ^{2}}{n ^{3}}

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