integral from 0 to x^2 of xcos(pix)

解

\int_{0}^{x^{2}} x cos (π x) d x

\frac{x ^{2} sin ( π x ^{2} )}{π} + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}

解答ステップ

\int_{0}^{x^{2}} x cos (π x) d x

部分積分を適用する

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

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

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

簡素化

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

限度を計算する:

\frac{1}{π} x^{2} sin (π x^{2}) + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}

= \frac{1}{π} x^{2} sin (π x^{2}) + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}

簡素化

= \frac{x ^{2} sin ( π x ^{2} )}{π} + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}