ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of 1/(1-csc(x))

解

\int \frac{1}{1 - csc ( x )} d x

解

- 4 (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2}))) + C

解答ステップ

\int \frac{1}{1 - csc ( x )} d x

u置換積分法を適用する

= \int \frac{4 u}{( 2 u - 1 - u ^{2} ) ( 1 + u ^{2} )} d u

定数を除く:

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

= 4 \cdot \int \frac{u}{( 2 u - 1 - u ^{2} ) ( 1 + u ^{2} )} d u

平方完成する

2 u - 1 - u^{2} : - (u - 1)^{2}

= 4 \cdot \int \frac{u}{( - ( u - 1 ) ^{2} ) ( 1 + u ^{2} )} d u

u置換積分法を適用する

= 4 \cdot \int - \frac{v + 1}{v ^{2} ( v ^{2} + 2 v + 2 )} d v

定数を除く:

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

= 4 (- \int \frac{v + 1}{v ^{2} ( v ^{2} + 2 v + 2 )} d v)

u置換積分法を適用する

= 4 (- \int \frac{w}{( w - 1 ) ^{2} ( w ^{2} + 1 )} d w)

以下の部分分数を得る:

\frac{w}{( w - 1 ) ^{2} ( w ^{2} + 1 )} : \frac{1}{2 ( w - 1 ) ^{2}} - \frac{1}{2 ( w ^{2} + 1 )}

= 4 (- \int \frac{1}{2 ( w - 1 ) ^{2}} - \frac{1}{2 ( w ^{2} + 1 )} d w)

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

= 4 (- (\int \frac{1}{2 ( w - 1 ) ^{2}} d w - \int \frac{1}{2 ( w ^{2} + 1 )} d w))

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

\int \frac{1}{2 ( w ^{2} + 1 )} d w = \frac{1}{2} arctan (w)

= 4 (- (- \frac{1}{2 ( w - 1 )} - \frac{1}{2} arctan (w)))

代用を戻す

= 4 (- (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 + 1 - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2}) - 1 + 1)))

簡素化

4 (- (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 + 1 - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2}) - 1 + 1))) : - 4 (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2})))

= - 4 (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2})))

定数を解答に追加する

= - 4 (- \frac{1}{2 ( tan ( \frac{x}{2} ) - 1 )} - \frac{1}{2} arctan (tan (\frac{x}{2}))) + C

グラフ

人気の例

積分記号 x*e^{2x}

in t e g r a l x \cdot e^{2 x}

integral of cot(1/2)x

\int cot (\frac{1}{2}) x d x

integral of rivative of x^2

\int \frac{d}{d x} (x^{2}) d x

integral of (2x+1)/(3x^2+8x-1)

\int \frac{2 x + 1}{3 x ^{2} + 8 x - 1} d x

integral of 1/(x(ln(x^2))^9)

\int \frac{1}{x ( ln ( x ^{2} ) ) ^{9}} d x