integral of (cos(x))^4

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

Vereinfache

cos^{4} (x) : cos^{3} (x) cos (x)

= \int cos^{3} (x) cos (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = \frac{x - 3}{x + 4}

l a pl a ce t r an s f or m t^{2} - 4 t + 4

x d x + y d y = x y (x d y - y d x)

\frac{d}{d x} (- 6 x^{\frac{1}{4}})

\int \frac{1}{x ( x - 1 )} d x