integral of cos(3x)cos(2x)cos(x)

Lösung

\int cos (3 x) cos (2 x) cos (x) d x

\frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x))) + C

Schritte zur Lösung

\int cos (3 x) cos (2 x) cos (x) d x

Wende die partielle Integration an

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

\int - \frac{1}{2} sin (x) (sin (x) + \frac{1}{5} sin (5 x)) d x = - \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x)))

= \frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) - (- \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x))))

Vereinfache

= \frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x))) + C

y^{^{'}} = \frac{( 6 x ^{4} e ^{\frac{y}{x}} + x ^{2} y ^{2} )}{x ^{3} y}

n = 0 \sum \infty \frac{1}{( n ^{2} - 1 )}

x \to 1 - lim (- \frac{2}{1 - x})

x \to 0 lim (ln (sin (x)))

so l v e f or x, y = sin (x y)