integral of (1-cos(x))/(sin(x))

Lösung

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

ln tan (\frac{x}{2}) - ln ∣ sin (x) ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{cos ( x )}{sin ( x )} d x = ln ∣ sin (x) ∣

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

Füge eine Konstante zur Lösung hinzu

= ln tan (\frac{x}{2}) - ln ∣ sin (x) ∣ + C

\int \frac{x}{( x ^{2} + a ^{2} ) ^{3}} d x

\int \frac{3}{7 - x ^{2}} d x

\int e^{3 x} (sin (2 x)) d x

\int tan^{2} (x) (x) d x

\int \frac{1}{t [ 1 + ( ln ( t ) ) ^{2} ]} d t