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Beliebt Rechnen >

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

Lösung

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

Lösung

x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x}{1 + sin ( x )} d x = x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣

\int \frac{cos ( x )}{1 + sin ( x )} d x = ln ∣ 1 + sin (x) ∣

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣

Füge eine Konstante zur Lösung hinzu

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣ + C

Graph

Beliebte Beispiele

integral of x^2\sqrt[3]{x^3+5}

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

integral of 1/(x(x^5+1))

\int \frac{1}{x ( x ^{5} + 1 )} d x

integral of (cos(sqrt(5x)))/(sqrt(5x))

\int \frac{cos ( 5 x )}{5 x} d x

integral of 1/((x-6)(x-5))

\int \frac{1}{( x - 6 ) ( x - 5 )} d x

integral of 1/x-5/(x^2+1)

\int \frac{1}{x} - \frac{5}{x ^{2} + 1} d x