integral of sin(4x)cos^2(2x)sin(x)

Lösung

\int sin (4 x) cos^{2} (2 x) sin (x) d x

- \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x) + C

Schritte zur Lösung

\int sin (4 x) cos^{2} (2 x) sin (x) d x

Wende die partielle Integration an

= \frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - \int - sin (4 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) d x

\int - sin (4 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) d x = - \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x)

= \frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - (- \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x))

Vereinfache

\frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - (- \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x)) : - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x)

= - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x) + C

d er i v a t i v e tan (3) x^{2}

d er i v a t i v e 3 t - 4 t^{2}

\frac{d y}{d x} y = cos^{- 1} (\frac{1}{x})

s l o p e (3.4) (- 6.4)

x \to 0 - lim (ln (x^{8}))