integral of (sin(pix)-pixcos(pix))/(x^2)

解

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

- \frac{sin ( π x )}{x} + C

解答ステップ

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

拡張

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

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{sin ( π x )}{x ^{2}} d x = - \frac{sin ( π x )}{x} + π Ci (π x)

\int \frac{π cos ( π x )}{x} d x = π Ci (π x)

= - \frac{sin ( π x )}{x} + π Ci (π x) - π Ci (π x)

簡素化

= - \frac{sin ( π x )}{x}

定数を解答に追加する

= - \frac{sin ( π x )}{x} + C