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人気のある微分積分 >

integral of (csc^4(x))/((1+cot(x))^2)

解

\int \frac{csc ^{4} ( x )}{( 1 + cot ( x ) ) ^{2}} d x

解

\frac{2}{cot ( x ) + 1} - cot (x) + 2 ln ∣ cot (x) + 1 ∣ + C

解答ステップ

\int \frac{csc ^{4} ( x )}{( 1 + cot ( x ) ) ^{2}} d x

u置換積分法を適用する

= \int - \frac{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

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

拡張

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

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

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

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

= - (- \frac{1}{1 + u} + u - 2 ln ∣ u + 1 ∣ - \frac{1}{u + 1})

代用を戻す

u = cot (x)

= - (- \frac{1}{1 + cot ( x )} + cot (x) - 2 ln ∣ cot (x) + 1 ∣ - \frac{1}{cot ( x ) + 1})

簡素化

- (- \frac{1}{1 + cot ( x )} + cot (x) - 2 ln ∣ cot (x) + 1 ∣ - \frac{1}{cot ( x ) + 1}) : \frac{2}{cot ( x ) + 1} - cot (x) + 2 ln ∣ cot (x) + 1 ∣

= \frac{2}{cot ( x ) + 1} - cot (x) + 2 ln ∣ cot (x) + 1 ∣

定数を解答に追加する

= \frac{2}{cot ( x ) + 1} - cot (x) + 2 ln ∣ cot (x) + 1 ∣ + C

グラフ

人気の例

integral of \sqrt[3]{6-2x}

\int 3 6 - 2 x d x

integral of (4x-7)/(sqrt(x^2+6x+5))

\int \frac{4 x - 7}{x ^{2} + 6 x + 5} d x

integral of x^2(9-x^2)

\int x^{2} (9 - x^{2}) d x

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

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

integral of (2x^4)/(x^2+3)

\int \frac{2 x ^{4}}{x ^{2} + 3} d x