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

解

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

\frac{16 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

解答ステップ

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

部分積分を適用する

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

\int \frac{4}{πn} x sin (\frac{πn x}{2}) d x = \frac{16}{π ^{3} n ^{3}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))

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

限度を計算する:

\frac{16 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= \frac{16 ( - 1 ) ^{n}}{π ^{2} n ^{2}}