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人気のある微分積分 >

integral of 1/(x(x^7+1)^2)

解

\int \frac{1}{x ( x ^{7} + 1 ) ^{2}} d x

解

\frac{1}{7} (ln x^{7} - ln x^{7} + 1 + \frac{1}{x ^{7} + 1}) + C

解答ステップ

\int \frac{1}{x ( x ^{7} + 1 ) ^{2}} d x

u置換積分法を適用する

= \int \frac{1}{7 u ( u + 1 ) ^{2}} d u

定数を除く:

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

= \frac{1}{7} \cdot \int \frac{1}{u ( u + 1 ) ^{2}} d u

u置換積分法を適用する

= \frac{1}{7} \cdot \int \frac{1}{( v - 1 ) v ^{2}} d v

以下の部分分数を得る:

\frac{1}{( v - 1 ) v ^{2}} : \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}}

= \frac{1}{7} \cdot \int \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}} d v

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

= \frac{1}{7} (\int \frac{1}{v - 1} d v - \int \frac{1}{v} d v - \int \frac{1}{v ^{2}} d v)

\int \frac{1}{v - 1} d v = ln ∣ v - 1 ∣

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

= \frac{1}{7} (ln ∣ v - 1 ∣ - ln ∣ v ∣ - (- \frac{1}{v}))

代用を戻す

= \frac{1}{7} (ln x^{7} + 1 - 1 - ln x^{7} + 1 - (- \frac{1}{x ^{7} + 1}))

簡素化

= \frac{1}{7} (ln x^{7} - ln x^{7} + 1 + \frac{1}{x ^{7} + 1})

定数を解答に追加する

= \frac{1}{7} (ln x^{7} - ln x^{7} + 1 + \frac{1}{x ^{7} + 1}) + C

グラフ

人気の例

derivative y=2*x^5-3*x*e^{-x}-10

d er i v a t i v e y = 2 \cdot x^{5} - 3 \cdot x \cdot e^{- x} - 10

limit as x approaches 2 of (x-4)/(x^2-6)

x \to 2 lim (\frac{x - 4}{x ^{2} - 6})

derivative of arctan(e^{x^2}+x^2)

\frac{d}{d x} (arctan (e^{x^{2}} + x^{2}))

(\partial)/(\partial x)(1/(x-y))

\frac{\partial}{\partial x} (\frac{1}{x - y})

integral of 1/(ax+b)

\int \frac{1}{a x + b} d x