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人気のある微分積分 >

integral of (1+4cot(x))/(4-cot(x))

解

\int \frac{1 + 4 cot ( x )}{4 - cot ( x )} d x

解

ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x)) + C

解答ステップ

\int \frac{1 + 4 cot ( x )}{4 - cot ( x )} d x

三角関数による置換を適用する

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

以下の部分分数を得る:

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

= - \int - \frac{1}{u - 4} + \frac{u}{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

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

\int \frac{1}{u - 4} d u = ln ∣ u - 4 ∣

\int \frac{u}{u ^{2} + 1} d u = \frac{1}{2} ln u^{2} + 1

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

代用を戻す

u = cot (x)

= - (- ln ∣ cot (x) - 4 ∣ + \frac{1}{2} ln cot^{2} (x) + 1)

簡素化

- (- ln ∣ cot (x) - 4 ∣ + \frac{1}{2} ln cot^{2} (x) + 1) : ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x))

= ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x))

定数を解答に追加する

= ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x)) + C

グラフ

人気の例

integral of x(2x+3)^7

\int x (2 x + 3)^{7} d x

integral of (cot(t)csc(t))/(csc^7(t))

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

integral of (2x-7)/(x^2-12x+32)

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

integral of ((4x^3))/(x^4+1)

\int \frac{( 4 x ^{3} )}{x ^{4} + 1} d x

integral of (xy^2+3x^2y)

\int (x y^{2} + 3 x^{2} y) d x