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Beliebt Rechnen >

integral of (sin(x)-cot(x))/(sin^2(x))

Lösung

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

Lösung

ln tan (\frac{x}{2}) + \frac{1}{2} csc^{2} (x) + C

Schritte zur Lösung

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

Multipliziere aus

\frac{sin ( x ) - cot ( x )}{sin ^{2} ( x )} : \frac{1}{sin ( x )} - \frac{cot ( x )}{sin ^{2} ( x )}

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

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

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

= ln tan (\frac{x}{2}) - (- \frac{1}{2} csc^{2} (x))

Vereinfache

= ln tan (\frac{x}{2}) + \frac{1}{2} csc^{2} (x)

Füge eine Konstante zur Lösung hinzu

= ln tan (\frac{x}{2}) + \frac{1}{2} csc^{2} (x) + C

Graph

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