de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= 22 \cdot \int \frac{2 u - 1}{2 u ^{2} + 1} d u

Multipliziere aus

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 22 (\int \frac{2 u}{2 u ^{2} + 1} d u - \int \frac{1}{2 u ^{2} + 1} d u)

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

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

= 22 (\frac{1}{2 2} ln 2 u^{2} + 1 - \frac{1}{2} arctan (2 u))

Setze in

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

ein

= 22 (\frac{1}{2 2} ln 2 (x + \frac{1}{2})^{2} + 1 - \frac{1}{2} arctan (2 (x + \frac{1}{2})))

Vereinfache

22 (\frac{1}{2 2} ln 2 (x + \frac{1}{2})^{2} + 1 - \frac{1}{2} arctan (2 (x + \frac{1}{2}))) : ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)

= ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)

Füge eine Konstante zur Lösung hinzu

= ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + C

Graph

Beliebte Beispiele

integral of (12x+5)

\int (12 x + 5) d x

(x^2+1)(dy)/(dx)+8x(y-1)=0

(x^{2} + 1) \frac{d y}{d x} + 8 x (y - 1) = 0

integral of (40x^2)/(x^4-52x^2+576)

\int \frac{40 x ^{2}}{x ^{4} - 52 x ^{2} + 576} d x

steigung (2,3),(-2,-3)

s l o p e (2, 3), (- 2, - 3)

derivative of 4/(2x+1)

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