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人気のある微分積分 >

integral of 2csc(x)(csc(x)-3cot(x))

解

\int 2 csc (x) (csc (x) - 3 cot (x)) d x

解

2 (- cot (x) + 3 csc (x)) + C

解答ステップ

\int 2 csc (x) (csc (x) - 3 cot (x)) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int csc (x) (csc (x) - 3 cot (x)) d x

三角関数の公式を使用して書き換える

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

拡張

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

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

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

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

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

= 2 (- cot (x) - (- 3 csc (x)))

簡素化

= 2 (- cot (x) + 3 csc (x))

定数を解答に追加する

= 2 (- cot (x) + 3 csc (x)) + C

グラフ

人気の例

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