integral of cot^6(x)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int - \frac{u ^{6}}{1 + u ^{2}} d u

Entferne die Konstante:

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

= - \int \frac{u ^{6}}{1 + u ^{2}} d u

Lange Division

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

= - \int u^{4} - u^{2} + 1 - \frac{1}{u ^{2} + 1} 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

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

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

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

\int 1 d u = u

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

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

Setze in

u = cot (x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

s l o p e (- 7.6) (4.2)

\frac{d}{d t} (\frac{t}{t - 1})

d er i v a t i v e 4 t e^{- t}

x \to - 1 lim (1 + e^{\frac{1}{x}})

a re a x^{2} - 3 x, 5 x, [0, 8]