integral of ((x^2cos(x)-2xsin(x)))/(x^4)

解

\int \frac{( x ^{2} cos ( x ) - 2 x sin ( x ) )}{x ^{4}} d x

\frac{sin ( x )}{x ^{2}} + C

解答ステップ

\int \frac{x ^{2} cos ( x ) - 2 x sin ( x )}{x ^{4}} d x

拡張

\frac{x ^{2} cos ( x ) - 2 x sin ( x )}{x ^{4}} : \frac{cos ( x )}{x ^{2}} - \frac{2 sin ( x )}{x ^{3}}

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

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

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

= - \frac{cos ( x )}{x} - Si (x) - (- \frac{sin ( x )}{x ^{2}} - \frac{cos ( x )}{x} - Si (x))

簡素化

= \frac{sin ( x )}{x ^{2}}

定数を解答に追加する

= \frac{sin ( x )}{x ^{2}} + C