de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of cot^6(3x)

Lösung

\int cot^{6} (3 x) d x

Lösung

- \frac{1}{3} (\frac{1}{5} cot^{5} (3 x) - \frac{1}{3} cot^{3} (3 x) + cot (3 x) - arctan (cot (3 x))) + C

Schritte zur Lösung

\int cot^{6} (3 x) d x

Wende U-Substitution an

= \int cot^{6} (u) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int cot^{6} (u) d u

Wende U-Substitution an

= \frac{1}{3} \cdot \int - \frac{v ^{6}}{1 + v ^{2}} d v

Entferne die Konstante:

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

= \frac{1}{3} (- \int \frac{v ^{6}}{1 + v ^{2}} d v)

Lange Division

\frac{v ^{6}}{1 + v ^{2}} : v^{4} - v^{2} + 1 - \frac{1}{v ^{2} + 1}

= \frac{1}{3} (- \int v^{4} - v^{2} + 1 - \frac{1}{v ^{2} + 1} d v)

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

\int v^{4} d v = \frac{v ^{5}}{5}

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

\int 1 d v = v

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{3} (- (\frac{v ^{5}}{5} - \frac{v ^{3}}{3} + v - arctan (v)))

Ersetze zurück

= \frac{1}{3} (- (\frac{cot ^{5} ( 3 x )}{5} - \frac{cot ^{3} ( 3 x )}{3} + cot (3 x) - arctan (cot (3 x))))

Vereinfache

\frac{1}{3} (- (\frac{cot ^{5} ( 3 x )}{5} - \frac{cot ^{3} ( 3 x )}{3} + cot (3 x) - arctan (cot (3 x)))) : - \frac{1}{3} (\frac{1}{5} cot^{5} (3 x) - \frac{1}{3} cot^{3} (3 x) + cot (3 x) - arctan (cot (3 x)))

= - \frac{1}{3} (\frac{1}{5} cot^{5} (3 x) - \frac{1}{3} cot^{3} (3 x) + cot (3 x) - arctan (cot (3 x)))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (\frac{1}{5} cot^{5} (3 x) - \frac{1}{3} cot^{3} (3 x) + cot (3 x) - arctan (cot (3 x))) + C

Graph

Beliebte Beispiele

integral of x^3sqrt(x+3)

\int x^{3} x + 3 d x

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

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

integral of sqrt((3x+2)/(x-3))

\int \frac{3 x + 2}{x - 3} d x

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

\int \frac{x + 2}{3 x} d x

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

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