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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

因数

2 x^{2} + 2 x + 1 : - (- 2 x^{2} - 2 x - 1)

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

定数を除く:

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

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

平方完成する

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

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

u置換積分法を適用する

= - \int \frac{2 u - 1}{- 4 u ^{2} - 1} d u

拡張

\frac{2 u - 1}{- 4 u ^{2} - 1} : \frac{2 u}{- 4 u ^{2} - 1} - \frac{1}{- 4 u ^{2} - 1}

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{2 u}{- 4 u ^{2} - 1} d u - \int \frac{1}{- 4 u ^{2} - 1} d u)

\int \frac{2 u}{- 4 u ^{2} - 1} d u = - \frac{1}{4} ln 4 u^{2} + 1

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

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

代用を戻す

u = x + \frac{1}{2}

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

簡素化

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

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

定数を解答に追加する

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

グラフ

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