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Beliebt Rechnen >

integral of 3((cos(x)+sin(x))/(sin(2x)))

Lösung

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

Lösung

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

Schritte zur Lösung

\int 3 \cdot \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

Multipliziere aus

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

= 3 \cdot \int \frac{cos ( x )}{sin ( 2 x )} + \frac{sin ( x )}{sin ( 2 x )} d x

Wende die Summenregel an:

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

= 3 (\int \frac{cos ( x )}{sin ( 2 x )} d x + \int \frac{sin ( x )}{sin ( 2 x )} d x)

\int \frac{cos ( x )}{sin ( 2 x )} d x = \frac{1}{2} ln tan (\frac{x}{2})

\int \frac{sin ( x )}{sin ( 2 x )} d x = \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from 2 to 6 of 4+2x^2

\int_{2}^{6} 4 + 2 x^{2} d x

integral of (10+x)^{-2}

\int (10 + x)^{- 2} d x

integral of (2x^4-x)/(x^3)

\int \frac{2 x ^{4} - x}{x ^{3}} d x

integral of (x+7)^7(x+8)

\int (x + 7)^{7} (x + 8) d x

derivative f(x)=(3x^2-x+1)/x

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