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

Lösung

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

\frac{d}{d x} (e^{- 8 x^{2}})

\frac{d}{d x} (\frac{3}{3 x - 5})

\int \frac{1}{5 x ^{2} - 20 x + 100} d x

\int \frac{5}{sec ( x ) - 1} d x

a re a x^{2} - 4 \cdot x, x \cdot 0, [- 2, 4]