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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1 + u ^{2}}{u} d u

Multipliziere aus

\frac{1 + u ^{2}}{u} : \frac{1}{u} + u

= \frac{1}{2} \cdot \int \frac{1}{u} + u d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= \frac{1}{2} (ln ∣ u ∣ + \frac{u ^{2}}{2})

Setze in

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

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x+3)(x+1)^{1/2}

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

integral of (4+6x)/(sqrt(x))

\int \frac{4 + 6 x}{x} d x

integral of (x^3)/(x^4+3)

\int \frac{x ^{3}}{x ^{4} + 3} d x

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\int x (x^{2} - 1)^{4} d x

integral of 3/(sqrt(9x^2+1))

\int \frac{3}{9 x ^{2} + 1} d x