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(2+x^{10))}

Lösung

\int \frac{1}{x ( 2 + x ^{10} )} d x

Lösung

\frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2) + C

Schritte zur Lösung

\int \frac{1}{x ( 2 + x ^{10} )} d x

Wende U-Substitution an

= \int \frac{1}{10 u ( 2 + u )} d u

Entferne die Konstante:

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

= \frac{1}{10} \cdot \int \frac{1}{u ( 2 + u )} d u

Ermittle den Partialbruch von

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

= \frac{1}{10} \cdot \int \frac{1}{2 u} - \frac{1}{2 ( u + 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

= \frac{1}{10} (\int \frac{1}{2 u} d u - \int \frac{1}{2 ( u + 2 )} d u)

\int \frac{1}{2 u} d u = \frac{1}{2} ln ∣ u ∣

\int \frac{1}{2 ( u + 2 )} d u = \frac{1}{2} ln ∣ u + 2 ∣

= \frac{1}{10} (\frac{1}{2} ln ∣ u ∣ - \frac{1}{2} ln ∣ u + 2 ∣)

Setze in

u = x^{10}

ein

= \frac{1}{10} (\frac{1}{2} ln x^{10} - \frac{1}{2} ln x^{10} + 2)

\frac{1}{2} ln x^{10} = 5 ln (x)

= \frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} (5 ln (x) - \frac{1}{2} ln x^{10} + 2) + C

Graph

Beliebte Beispiele

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

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

integral of 8^xsin(8^x)

\int 8^{x} sin (8^{x}) d x

integral of 1/(3\sqrt[3]{x)}

\int \frac{1}{3 3 x} d x

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

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

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

\int \frac{cos ( x )}{2 sin ( x ) + 3} d x