integral of 5(cos(x)+sin(x))/(sin(2x))

Lösung

\int 5 \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

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

Schritte zur Lösung

\int 5 \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

= 5 \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 )}

= 5 \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

= 5 (\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) ∣

= 5 (\frac{1}{2} ln tan (\frac{x}{2}) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{1}{2} ln tan (\frac{x}{2}) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

d er i v a t i v e f (x) = \frac{x}{x ^{2} + 81}

\int_{0}^{1} (1 - x^{3}) d x

\frac{d}{d x} ((tan (2 x))^{3})

d er i v a t i v e (2 x - 7)^{4} (x^{2} + x + 1)^{5}

d er i v a t i v e (9 z^{4} + 3 z^{2} + 1) (3 z^{3} - z)