integral (sin^2(x))^2

Lösung

integral

(sin^{2} (x))^{2}

- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) + C

Schritte zur Lösung

\int (sin^{2} (x))^{2} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int sin^{4} (x) d x

Wende integrale Reduktion an

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

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

= - \frac{cos ( x ) sin ^{3} ( x )}{4} + \frac{3}{4} \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x))

Vereinfache

- \frac{cos ( x ) sin ^{3} ( x )}{4} + \frac{3}{4} \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

= - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) + C

in t e g r a l \frac{1}{( 1 + sin ^{2} ( x ) )}

in t e g r a l x - 2 sin (x)

in t e g r a l \frac{cos ^{2} ( t )}{2}

in t e g r a l \frac{x}{( x + 1 )}

in t e g r a l 3 - 2 x