de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x^2sqrt(a^2+x^2)

\int x^{2} a^{2} + x^{2} d x

integral of 1/(sqrt(1-16y^2))

\int \frac{1}{1 - 16 y ^{2}} d y

integral of e^{2y}(4-e^{2y})^{4/3}

\int e^{2 y} (4 - e^{2 y})^{\frac{4}{3}} d y

integral of (e^{y^2}y)

\int (e^{y^{2}} y) d y

integral of (-3x^2)/(sqrt(16-x^2))

\int \frac{- 3 x ^{2}}{16 - x ^{2}} d x