integral of csc^4(x)cot^2(x)

Lösung

\int csc^{4} (x) cot^{2} (x) d x

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

Schritte zur Lösung

\int csc^{4} (x) cot^{2} (x) d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Multipliziere aus

u^{2} (u^{2} + 1) : u^{4} + u^{2}

= - \int u^{4} + 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

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

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

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

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

Setze in

u = cot (x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to 2 lim (5 x^{3} + x)

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

x \to 0 + lim (ln (1))

d er i v a t i v e \frac{( 4 x ^{5} + 4 x ^{4} - 3 x ^{3} )}{x ^{4}}

d er i v a t i v e y = 4 + x - 9