integral from 0 to 4 of x^2cos((npix)/4)

解

\int_{0}^{4} x^{2} cos (\frac{nπ x}{4}) d x

\frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

解答ステップ

\int_{0}^{4} x^{2} cos (\frac{nπ x}{4}) d x

部分積分を適用する

= [\frac{4}{πn} x^{2} sin (\frac{πn x}{4}) - \int \frac{8}{πn} x sin (\frac{πn x}{4}) d x]_{0}^{4}

\int \frac{8}{πn} x sin (\frac{πn x}{4}) d x = \frac{8}{π ^{3} n ^{3}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4}))

= [\frac{4}{πn} x^{2} sin (\frac{πn x}{4}) - \frac{8}{π ^{3} n ^{3}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4}))]_{0}^{4}

限度を計算する:

\frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= \frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}