integral of (cos^5(x))/(sin^3(x))

Lösung

\int \frac{cos ^{5} ( x )}{sin ^{3} ( x )} d x

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

Schritte zur Lösung

\int \frac{cos ^{5} ( x )}{sin ^{3} ( x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 1 - sin ^{2} ( x ) ) ^{2} cos ( x )}{sin ^{3} ( x )} d x

Wende U-Substitution an

= \int \frac{( 1 - u ^{2} ) ^{2}}{u ^{3}} d u

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{2}}{u ^{3}} : \frac{1}{u ^{3}} - \frac{2}{u} + u

= \int \frac{1}{u ^{3}} - \frac{2}{u} + u d u

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}{u ^{3}} d u - \int \frac{2}{u} d u + \int u d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

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

Setze in

u = sin (x)

ein

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

Vereinfache

- \frac{1}{2 sin ^{2} ( x )} - 2 ln ∣ sin (x) ∣ + \frac{sin ^{2} ( x )}{2} : - \frac{1}{2} csc^{2} (x) - 2 ln ∣ sin (x) ∣ + \frac{1}{2} sin^{2} (x)

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

Füge eine Konstante zur Lösung hinzu

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

\int_{0}^{0.6} \frac{1}{( 1 - x ) ( 3 - x )} d x

\int \frac{1}{t ^{2} 2 - t ^{2}} d t

d er i v a t i v e \frac{( cos ( x ) )}{cos ^{2} ( \cdot x ) - 1}

\int (n + 3) 3 d n

\int \frac{2}{e ^{x} + 2} d x