integral of 1/((2+cos(x))sin(x))

Lösung

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

\frac{1}{3} ln tan (\frac{x}{2}) + \frac{1}{3} ln tan^{2} (\frac{x}{2}) + 3 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{1 + u ^{2}}{u ( u ^{2} + 3 )} d u

Multipliziere aus

\frac{1 + u ^{2}}{u ( u ^{2} + 3 )} : \frac{1}{u ( u ^{2} + 3 )} + \frac{u ^{2}}{u ( u ^{2} + 3 )}

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

\int \frac{1}{u ( u ^{2} + 3 )} d u = \frac{1}{3} ln ∣ u ∣ - \frac{1}{6} ln u^{2} + 3

\int \frac{u ^{2}}{u ( u ^{2} + 3 )} d u = \frac{1}{2} ln u^{2} + 3

= \frac{1}{3} ln ∣ u ∣ - \frac{1}{6} ln u^{2} + 3 + \frac{1}{2} ln u^{2} + 3

Setze in

u = tan (\frac{x}{2})

ein

= \frac{1}{3} ln tan (\frac{x}{2}) - \frac{1}{6} ln tan^{2} (\frac{x}{2}) + 3 + \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 3

Addiere gleiche Elemente:

- \frac{1}{6} ln tan^{2} (\frac{x}{2}) + 3 + \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 3 = \frac{1}{3} ln tan^{2} (\frac{x}{2}) + 3

= \frac{1}{3} ln tan (\frac{x}{2}) + \frac{1}{3} ln tan^{2} (\frac{x}{2}) + 3

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln tan (\frac{x}{2}) + \frac{1}{3} ln tan^{2} (\frac{x}{2}) + 3 + C

\int e^{2 x - 1} d x

\int v^{2} e^{- v^{2}} d v

\int x^{2} sin^{3} (x^{3}) cos^{2} (x^{3}) d x

in t e g r a l 1

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