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Beliebt Rechnen >

integral of (2-sin(x))/(2+cos(x))

Lösung

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

Lösung

\frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{2 - sin ( x )}{2 + cos ( x )} : \frac{2}{2 + cos ( x )} - \frac{sin ( x )}{2 + cos ( 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}{2 + cos ( x )} d x - \int \frac{sin ( x )}{2 + cos ( x )} d x

\int \frac{2}{2 + cos ( x )} d x = \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

\int \frac{sin ( x )}{2 + cos ( x )} d x = - ln ∣ 2 + cos (x) ∣

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - (- ln ∣ 2 + cos (x) ∣)

Vereinfache

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

Graph

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derivative cos(3x)

d er i v a t i v e cos (3 x)

\int 60 d x

integral of x^4sqrt(x+1)

\int x^{4} x + 1 d x

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d er i v a t i v e f (x) = (3 x + 1)^{2}

integral from 0 to 2 of integral from 3y to 6 of 5e^{x^2}

\int_{0}^{2} \int_{3 y}^{6} 5 e^{x^{2}} d x d y