integral of (sin(x))^6

Lösung

\int (sin (x))^{6} d x

- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \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))^{6} d x

Vereinfache

= \int sin^{6} (x) d x

Wende integrale Reduktion an

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} \cdot \int sin^{4} (x) d x

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

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \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{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) + C

\frac{\partial}{\partial x} (z e^{x})

\int 4 e^{x} + cos (\frac{x}{2}) d x

\frac{d}{d x} ((- 25 x^{2} + 1)^{2})

\frac{\partial}{\partial x} (- \frac{7 x ^{2}}{( x - y ) ^{2}})

\frac{d}{d x} (a x^{b^{2} - c})