integral of (cos(u))/(sin(u)+sin^3(u))

Lösung

\int \frac{cos ( u )}{sin ( u ) + sin ^{3} ( u )} d u

ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1 + C

Schritte zur Lösung

\int \frac{cos ( u )}{sin ( u ) + sin ^{3} ( u )} d u

Wende U-Substitution an

= \int \frac{1}{v + v ^{3}} d v

Ermittle den Partialbruch von

\frac{1}{v + v ^{3}} : \frac{1}{v} - \frac{v}{v ^{2} + 1}

= \int \frac{1}{v} - \frac{v}{v ^{2} + 1} d v

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}{v} d v - \int \frac{v}{v ^{2} + 1} d v

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{v}{v ^{2} + 1} d v = \frac{1}{2} ln v^{2} + 1

= ln ∣ v ∣ - \frac{1}{2} ln v^{2} + 1

Setze in

v = sin (u)

ein

= ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1

Füge eine Konstante zur Lösung hinzu

= ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1 + C

\int e^{\frac{- x ^{2}}{5}} d x

\int 12 t e^{t} d t

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

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

\int 2 x y^{2} + 2 y d x