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

Lösung

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

2 csc (x) - 2 cot (x) - x + C

Schritte zur Lösung

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

\frac{1 - cos ( x )}{1 + cos ( x )} = - \frac{2 cos ( x )}{1 + cos ( x )} + 1

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

\int \frac{2 cos ( x )}{1 + cos ( x )} d x = 2 (- csc (x) + cot (x) + x)

\int 1 d x = x

= - 2 (- csc (x) + cot (x) + x) + x

Vereinfache

- 2 (- csc (x) + cot (x) + x) + x : 2 csc (x) - 2 cot (x) - x

= 2 csc (x) - 2 cot (x) - x

Füge eine Konstante zur Lösung hinzu

= 2 csc (x) - 2 cot (x) - x + C

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