de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 1/(x^5+x^4)

Lösung

\int \frac{1}{x ^{5} + x ^{4}} d x

Lösung

- ln ∣ x ∣ - \frac{1}{x} + \frac{1}{2 x ^{2}} - \frac{1}{3 x ^{3}} + ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{5} + x ^{4}} d x

Faktorisiere

\frac{1}{x ^{5} + x ^{4}}

= \int \frac{1}{x ^{4} ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{1}{x ^{4} ( x + 1 )} : - \frac{1}{x} + \frac{1}{x ^{2}} - \frac{1}{x ^{3}} + \frac{1}{x ^{4}} + \frac{1}{x + 1}

= \int - \frac{1}{x} + \frac{1}{x ^{2}} - \frac{1}{x ^{3}} + \frac{1}{x ^{4}} + \frac{1}{x + 1} 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{1}{x} d x + \int \frac{1}{x ^{2}} d x - \int \frac{1}{x ^{3}} d x + \int \frac{1}{x ^{4}} d x + \int \frac{1}{x + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

= - ln ∣ x ∣ - \frac{1}{x} - (- \frac{1}{2 x ^{2}}) - \frac{1}{3 x ^{3}} + ln ∣ x + 1 ∣

Vereinfache

= - ln ∣ x ∣ - \frac{1}{x} + \frac{1}{2 x ^{2}} - \frac{1}{3 x ^{3}} + ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x ∣ - \frac{1}{x} + \frac{1}{2 x ^{2}} - \frac{1}{3 x ^{3}} + ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of 7cos(7x)

integral of x^3sin(ax)

integral of-3x^8+4/(x-7)

integral of cos(x)(2+sin(x))^3

integral of e^{-2at}