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^3-1)/((4x^2-1)^2)

Lösung

\int \frac{x ^{3} - 1}{( 4 x ^{2} - 1 ) ^{2}} d x

Lösung

- \frac{3}{32} ln ∣ 2 x + 1 ∣ + \frac{9}{64 ( 2 x + 1 )} + \frac{5}{32} ln ∣ 2 x - 1 ∣ + \frac{7}{64 ( 2 x - 1 )} + C

Schritte zur Lösung

\int \frac{x ^{3} - 1}{( 4 x ^{2} - 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{3} - 1}{( 4 x ^{2} - 1 ) ^{2}} : - \frac{3}{16 ( 2 x + 1 )} - \frac{9}{32 ( 2 x + 1 ) ^{2}} + \frac{5}{16 ( 2 x - 1 )} - \frac{7}{32 ( 2 x - 1 ) ^{2}}

= \int - \frac{3}{16 ( 2 x + 1 )} - \frac{9}{32 ( 2 x + 1 ) ^{2}} + \frac{5}{16 ( 2 x - 1 )} - \frac{7}{32 ( 2 x - 1 ) ^{2}} d x

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{3}{16 ( 2 x + 1 )} d x - \int \frac{9}{32 ( 2 x + 1 ) ^{2}} d x + \int \frac{5}{16 ( 2 x - 1 )} d x - \int \frac{7}{32 ( 2 x - 1 ) ^{2}} d x

\int \frac{3}{16 ( 2 x + 1 )} d x = \frac{3}{32} ln ∣ 2 x + 1 ∣

\int \frac{9}{32 ( 2 x + 1 ) ^{2}} d x = - \frac{9}{64 ( 2 x + 1 )}

\int \frac{5}{16 ( 2 x - 1 )} d x = \frac{5}{32} ln ∣ 2 x - 1 ∣

\int \frac{7}{32 ( 2 x - 1 ) ^{2}} d x = - \frac{7}{64 ( 2 x - 1 )}

= - \frac{3}{32} ln ∣ 2 x + 1 ∣ - (- \frac{9}{64 ( 2 x + 1 )}) + \frac{5}{32} ln ∣ 2 x - 1 ∣ - (- \frac{7}{64 ( 2 x - 1 )})

Vereinfache

= - \frac{3}{32} ln ∣ 2 x + 1 ∣ + \frac{9}{64 ( 2 x + 1 )} + \frac{5}{32} ln ∣ 2 x - 1 ∣ + \frac{7}{64 ( 2 x - 1 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{32} ln ∣ 2 x + 1 ∣ + \frac{9}{64 ( 2 x + 1 )} + \frac{5}{32} ln ∣ 2 x - 1 ∣ + \frac{7}{64 ( 2 x - 1 )} + C

Graph

Beliebte Beispiele

integral of e^{-x}sin(2x+1)

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

integral of 1/(2-3e^{-x)}

\int \frac{1}{2 - 3 e ^{- x}} d x

integral 1/(sqrt(x^2-1))

in t e g r a l \frac{1}{x ^{2} - 1}

integral of-cos(x)2

\int - cos (x) 2 d x

integral of s/(sqrt(s+1))

\int \frac{s}{s + 1} d s