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Beliebt Rechnen >

integral of 1/(4-5sin(x))

Lösung

\int \frac{1}{4 - 5 sin ( x )} d x

Lösung

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

Schritte zur Lösung

\int \frac{1}{4 - 5 sin ( x )} d x

Wende U-Substitution an

= \int \frac{1}{2 ( u ^{2} + 1 ) - 5 u} d u

Ermittle den Partialbruch von

\frac{1}{2 ( u ^{2} + 1 ) - 5 u} : - \frac{2}{3 ( 2 u - 1 )} + \frac{1}{3 ( u - 2 )}

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

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

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

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

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

Setze in

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

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (5x+3)/((x+2)^2(x-1)^2)

\int \frac{5 x + 3}{( x + 2 ) ^{2} ( x - 1 ) ^{2}} d x

integral of sec^2(2x)tan(2x)

\int sec^{2} (2 x) tan (2 x) d x

integral of 1/(1+sqrt(2)x)

\int \frac{1}{1 + 2 x} d x

integral of ((7x-1))/((x-1)(x-2)(x-3))

\int \frac{( 7 x - 1 )}{( x - 1 ) ( x - 2 ) ( x - 3 )} d x

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

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