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 (x^2+x+2)/(x^4+5x^2+4)

解

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

解

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

解答ステップ

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

因数

\frac{x ^{2} + x + 2}{x ^{4} + 5 x ^{2} + 4}

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

拡張

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

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

\int \frac{x ^{2}}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x = - \frac{1}{3} arctan (x) + \frac{2}{3} arctan (\frac{x}{2})

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

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

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of 6x^5-8x^4-9x^2

\int 6 x^{5} - 8 x^{4} - 9 x^{2} d x

y^{^{'}} = 2 t e^{7 y}

integral of xe^{-4x}

\int x e^{- 4 x} d x

derivative f(x)=(-3)/(x^4)

d er i v a t i v e f (x) = \frac{- 3}{x ^{4}}

integral from 4 to 69 of 1/(x^3-64)

\int_{4}^{69} \frac{1}{x ^{3} - 64} d x