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人気のある微分積分 >

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

解

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

解

\frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2})) + C

解答ステップ

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

三角関数の公式を使用して書き換える

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

以下の部分分数を得る:

\frac{u}{( 2 u - 1 + u ^{2} ) ( 1 + u ^{2} )} : \frac{u + 1}{4 ( u ^{2} + 2 u - 1 )} + \frac{- u + 1}{4 ( u ^{2} + 1 )}

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

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

\int \frac{u + 1}{4 ( u ^{2} + 2 u - 1 )} d u = \frac{1}{8} ln u^{2} + 2 u - 1

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

= 4 (\frac{1}{8} ln u^{2} + 2 u - 1 + \frac{1}{4} (- \frac{1}{2} ln u^{2} + 1 + arctan (u)))

代用を戻す

u = tan (\frac{x}{2})

= 4 (\frac{1}{8} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 + \frac{1}{4} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 + arctan (tan (\frac{x}{2}))))

簡素化

4 (\frac{1}{8} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 + \frac{1}{4} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 + arctan (tan (\frac{x}{2})))) : \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2}))

= \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2}))

定数を解答に追加する

= \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2})) + C

グラフ

人気の例

integral of (x^2+1)^{(-1)/2}

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

integral of sqrt(1+x^{(-2)/3)}

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

integral of (1/(5x))

\int (\frac{1}{5 x}) d x

integral of (x/(x^2))

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

integral of sinh^2(t)

\int sinh^{2} (t) d t