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

Lösung

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

\frac{2}{3} sin^{3} (x) cos (3 x) + 2 (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int - 2 sin^{3} (x) sin (3 x) d x = - 2 (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x))

= \frac{2}{3} sin^{3} (x) cos (3 x) - (- 2 (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)))

Vereinfache

= \frac{2}{3} sin^{3} (x) cos (3 x) + 2 (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x))

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} sin^{3} (x) cos (3 x) + 2 (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)) + C

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

x \to 3 + lim (\frac{3 - x}{x ^{2} - 9})

\frac{d ^{2}}{d x ^{2}} f (x) = x^{3} + 3 x^{2} - 2 x + 1

\frac{\partial}{\partial x} (2 x y e^{x z})

ma c l a u r in x