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人気のある微分積分 >

integral of 1/(x^7+4x)

解

\int \frac{1}{x ^{7} + 4 x} d x

解

\frac{1}{6} (\frac{3}{2} ln (x) - \frac{1}{4} ln x^{6} + 4) + C

解答ステップ

\int \frac{1}{x ^{7} + 4 x} d x

因数

x^{7} + 4 x : x (x^{6} + 4)

= \int \frac{1}{x ( x ^{6} + 4 )} d x

u置換積分法を適用する

= \int \frac{1}{6 u ( u + 4 )} d u

定数を除く:

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

= \frac{1}{6} \cdot \int \frac{1}{u ( u + 4 )} d u

以下の部分分数を得る:

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

= \frac{1}{6} \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 4 )} 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

= \frac{1}{6} (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 4 )} d u)

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

\int \frac{1}{4 ( u + 4 )} d u = \frac{1}{4} ln ∣ u + 4 ∣

= \frac{1}{6} (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 4 ∣)

代用を戻す

u = x^{6}

= \frac{1}{6} (\frac{1}{4} ln x^{6} - \frac{1}{4} ln x^{6} + 4)

\frac{1}{4} ln x^{6} = \frac{3}{2} ln (x)

= \frac{1}{6} (\frac{3}{2} ln (x) - \frac{1}{4} ln x^{6} + 4)

定数を解答に追加する

= \frac{1}{6} (\frac{3}{2} ln (x) - \frac{1}{4} ln x^{6} + 4) + C

グラフ

人気の例

integral of (4x-5)^{1/2}

\int (4 x - 5)^{\frac{1}{2}} d x

integral of pi/2 sin(pi/6 x)

\int \frac{π}{2} sin (\frac{π}{6} x) d x

integral of (2x)/(x-2)

\int \frac{2 x}{x - 2} d x

integral of 1/(81+36x^2)

\int \frac{1}{81 + 36 x ^{2}} d x

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

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