ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of (2x^4)/(x^3-x^2+x-1)

解

\int \frac{2 x ^{4}}{x ^{3} - x ^{2} + x - 1} d x

解

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

解答ステップ

\int \frac{2 x ^{4}}{x ^{3} - x ^{2} + x - 1} d x

定数を除く:

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

= 2 \cdot \int \frac{x ^{4}}{x ^{3} - x ^{2} + x - 1} d x

長除法

\frac{x ^{4}}{x ^{3} - x ^{2} + x - 1} : x + 1 + \frac{1}{x ^{3} - x ^{2} + x - 1}

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

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

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

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

\int \frac{1}{x ^{3} - x ^{2} + x - 1} d x = \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x))

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of (sqrt(1-25x^2))/(x^2)

\int \frac{1 - 25 x ^{2}}{x ^{2}} d x

integral of 3(cos(x)+sin(x))/(sin(2x))

\int 3 \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

derivative e^{2x+1}

d er i v a t i v e e^{2 x + 1}

derivative f(x)=5x^{-2}

d er i v a t i v e f (x) = 5 x^{- 2}

integral of (3x-2)e-x

\int (3 x - 2) e - x d x