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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of e^{9x}sin(2x)

\int e^{9 x} sin (2 x) d x

integral of cos(x)(sqrt((9+25sin^2(x))))

\int cos (x) ((9 + 25 sin^{2} (x))) d x

integral of 2/(x+4)

\int \frac{2}{x + 4} d x

integral of (x^2+2x+5)/(x+4)

\int \frac{x ^{2} + 2 x + 5}{x + 4} d x

integral of sin^6(x/2)cos^5(x/2)

\int sin^{6} (\frac{x}{2}) cos^{5} (\frac{x}{2}) d x