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^{11})/(sqrt(x^6-1))

Lösung

\int \frac{x ^{11}}{x ^{6} - 1} d x

Lösung

\frac{1}{9} (x^{6} + 2) x^{6} - 1 + C

Schritte zur Lösung

\int \frac{x ^{11}}{x ^{6} - 1} d x

Wende U-Substitution an

= \int \frac{u + 1}{6 u} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{u + 1}{u} d u

Multipliziere aus

\frac{u + 1}{u} : u + \frac{1}{u}

= \frac{1}{6} \cdot \int u + \frac{1}{u} 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}{6} (\int u d u + \int \frac{1}{u} d u)

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{1}{6} (\frac{2}{3} u^{\frac{3}{2}} + 2 u)

Setze in

u = x^{6} - 1

ein

= \frac{1}{6} (\frac{2}{3} (x^{6} - 1)^{\frac{3}{2}} + 2 x^{6} - 1)

Vereinfache

\frac{1}{6} (\frac{2}{3} (x^{6} - 1)^{\frac{3}{2}} + 2 x^{6} - 1) : \frac{1}{9} (x^{6} + 2) x^{6} - 1

= \frac{1}{9} (x^{6} + 2) x^{6} - 1

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (x^{6} + 2) x^{6} - 1 + C

Graph

Beliebte Beispiele

integral of 16x^{-3}

\int 16 x^{- 3} d x

integral of (t^2)/(sqrt(t^3-1))

\int \frac{t ^{2}}{t ^{3} - 1} d t

integral of (x^3)/((1+x))

\int \frac{x ^{3}}{( 1 + x )} d x

integral of x^{1.4}+9x^{3.5}

\int x^{1.4} + 9 x^{3.5} d x

integral of y/((1+y^2)^2)

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