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

Lösung

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

- cot (x) + ln (sin^{2} (x)) + C

Schritte zur Lösung

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

Multipliziere aus

\frac{1 + sin ( 2 x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} + \frac{sin ( 2 x )}{sin ^{2} ( x )}

= \int \frac{1}{sin ^{2} ( x )} + \frac{sin ( 2 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

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

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

\int \frac{sin ( 2 x )}{sin ^{2} ( x )} d x = ln (sin^{2} (x))

= - cot (x) + ln (sin^{2} (x))

Füge eine Konstante zur Lösung hinzu

= - cot (x) + ln (sin^{2} (x)) + C

\int - x + x^{3} d x

\int (x^{2} - 9) 2 x d x

\int \frac{1}{16 + 4 x ^{2}} d x

\int sin (x) (sec (x))^{3} d x

\int e^{- 2} sin (\frac{x}{2}) d x