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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

平方完成する

x^{2} + 2 x + 4 : (x + 1)^{2} + 3

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

u置換積分法を適用する

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

拡張

\frac{u - 1}{u ^{2} + 3} : \frac{u}{u ^{2} + 3} - \frac{1}{u ^{2} + 3}

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

\int \frac{u}{u ^{2} + 3} d u = \frac{1}{2} ln u^{2} + 3

\int \frac{1}{u ^{2} + 3} d u = \frac{1}{3} arctan (\frac{u}{3})

= \frac{1}{2} ln u^{2} + 3 - \frac{1}{3} arctan (\frac{u}{3})

代用を戻す

u = x + 1

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

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