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

Lösung

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

\frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

Schritte zur Lösung

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

Multipliziere mit der Konjugation von

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

= \int \frac{sin ( x )}{( \frac{c o s ^{2} ( x )}{1 + s i n ( x )} ) ^{2}} d x

Multipliziere aus

\frac{sin ( x )}{( \frac{c o s ^{2} ( x )}{1 + s i n ( x )} ) ^{2}} : \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( x )}

= \int \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( 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{sin ( x )}{cos ^{4} ( x )} d x + \int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x + \int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x

\int \frac{sin ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x)

\int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x = \frac{2}{3} tan^{3} (x)

\int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x) - sec (x)

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

Addiere gleiche Elemente:

\frac{1}{3} sec^{3} (x) + \frac{1}{3} sec^{3} (x) = \frac{2}{3} sec^{3} (x)

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x)

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

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

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

\int \frac{x ^{2} + 6 x - 1}{x ^{3} - x} d x

\int \frac{1}{1 - y} d y

\int \frac{- 8 x}{( x + 1 ) - x + 1} d x