積分記号 (1/(x^4))sin(x)

解

積分記号

(\frac{1}{x ^{4}}) sin (x)

- \frac{sin ( x )}{3 x ^{3}} + \frac{1}{3} (- \frac{cos ( x )}{2 x ^{2}} - \frac{1}{2} (- \frac{sin ( x )}{x} + Ci (x))) + C

解答ステップ

\int \frac{1}{x ^{4}} sin (x) d x

簡素化

\frac{1}{x ^{4}} sin (x) : \frac{sin ( x )}{x ^{4}}

= \int \frac{sin ( x )}{x ^{4}} d x

部分積分を適用する

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

\int - \frac{cos ( x )}{3 x ^{3}} d x = - \frac{1}{3} (- \frac{cos ( x )}{2 x ^{2}} - \frac{1}{2} (- \frac{sin ( x )}{x} + Ci (x)))

= - \frac{sin ( x )}{3 x ^{3}} - (- \frac{1}{3} (- \frac{cos ( x )}{2 x ^{2}} - \frac{1}{2} (- \frac{sin ( x )}{x} + Ci (x))))

簡素化

= - \frac{sin ( x )}{3 x ^{3}} + \frac{1}{3} (- \frac{cos ( x )}{2 x ^{2}} - \frac{1}{2} (- \frac{sin ( x )}{x} + Ci (x)))

定数を解答に追加する

= - \frac{sin ( x )}{3 x ^{3}} + \frac{1}{3} (- \frac{cos ( x )}{2 x ^{2}} - \frac{1}{2} (- \frac{sin ( x )}{x} + Ci (x))) + C