integral of (sin(x))^4

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

= \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

\frac{d}{d x} (5 x - 5 e^{x})

t a y l or x^{(\frac{1}{3})}, 8

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

\int_{1}^{x} 1 + 2 t d t

\frac{d}{d x} (\frac{4}{3} \cdot π \cdot x^{3})