integral of (1-sin^3(x))/(sin^2(x))

解

\int \frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} d x

- cot (x) + cos (x) + C

解答ステップ

\int \frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} d x

拡張

\frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - sin (x)

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

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

\int sin (x) d x = - cos (x)

= - cot (x) - (- cos (x))

簡素化

= - cot (x) + cos (x)

定数を解答に追加する

= - cot (x) + cos (x) + C