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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere mit der Konjugation von

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

= \int \frac{cos ( x )}{\frac{s i n ^{2} ( x )}{1 - c o s ( x )}} d x

Multipliziere aus

\frac{cos ( x )}{\frac{s i n ^{2} ( x )}{1 - c o s ( x )}} : \frac{cos ( x )}{sin ^{2} ( x )} - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

(dy)/(dx)=(1-x^2)/(y^2)

\frac{d y}{d x} = \frac{1 - x ^{2}}{y ^{2}}

derivative sqrt(t+t^3)

d er i v a t i v e t + t^{3}

\frac{d y}{d x} = 8 - 4 y

integral of (e^{2sqrt(x)})/(3sqrt(x))

\int \frac{e ^{2 x}}{3 x} d x

derivative of x^3-4x+7

\frac{d}{d x} (x^{3} - 4 x + 7)