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

Lösung

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

Lösung

- \frac{1}{2 ( x + 1 )} - \frac{1}{2 ( x - 1 )} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

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

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

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

= - \frac{1}{2 u} - \frac{1}{2 ( u - 2 )}

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x + 1 )} - \frac{1}{2 ( x - 1 )} + C

Graph

Beliebte Beispiele

integral of sin((npi)/l x)

\int sin (\frac{nπ}{l} x) d x

integral of 1/(6\sqrt[6]{x^5)}

\int \frac{1}{6 6 x ^{5}} d x

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

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

integral of (e^{2x}\sqrt[3]{e^x+2})

\int (e^{2 x} 3 e^{x} + 2) d x

integral of-2/(1+x)

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