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Beliebt Rechnen >

integral of (2-sqrt(cot(x)))^2csc^2(x)

Lösung

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

Lösung

- 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 2 - cot ( x ) ) ^{2}}{sin ^{2} ( x )} d x

Multipliziere aus

\frac{( 2 - cot ( x ) ) ^{2}}{sin ^{2} ( x )} : \frac{4}{sin ^{2} ( x )} - \frac{4 cot ( x )}{sin ^{2} ( x )} + \frac{cot ( x )}{sin ^{2} ( x )}

= \int \frac{4}{sin ^{2} ( x )} - \frac{4 cot ( x )}{sin ^{2} ( x )} + \frac{cot ( x )}{sin ^{2} ( x )} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4}{sin ^{2} ( x )} d x - \int \frac{4 cot ( x )}{sin ^{2} ( x )} d x + \int \frac{cot ( x )}{sin ^{2} ( x )} d x

\int \frac{4}{sin ^{2} ( x )} d x = - 4 cot (x)

\int \frac{4 cot ( x )}{sin ^{2} ( x )} d x = - \frac{8}{3} cot^{\frac{3}{2}} (x)

\int \frac{cot ( x )}{sin ^{2} ( x )} d x = - \frac{1}{2} csc^{2} (x)

= - 4 cot (x) - (- \frac{8}{3} cot^{\frac{3}{2}} (x)) - \frac{1}{2} csc^{2} (x)

Vereinfache

= - 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x)

Füge eine Konstante zur Lösung hinzu

= - 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x) + C

Graph

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