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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

integral of sqrt(x)(2+x)

\int x (2 + x) d x

integral of x^2+5x-3

\int x^{2} + 5 x - 3 d x

integral of (x^2+1)/((2x-1)^3)

\int \frac{x ^{2} + 1}{( 2 x - 1 ) ^{3}} d x

integral of (x^3+3x)^8(x^2+1)

\int (x^{3} + 3 x)^{8} (x^{2} + 1) d x

integral of-e^xsin(e^x)

\int - e^{x} sin (e^{x}) d x