integral of (x^2+5x+6)cos(2x)

Lösung

\int (x^{2} + 5 x + 6) cos (2 x) d x

\frac{1}{2} x^{2} sin (2 x) + \frac{5}{2} x sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{11}{4} sin (2 x) + \frac{5}{4} cos (2 x) + C

Schritte zur Lösung

\int (x^{2} + 5 x + 6) cos (2 x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

\frac{1}{2} sin (2 x) (x^{2} + 5 x + 6) - (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x) - \frac{5}{4} cos (2 x)) : \frac{1}{2} x^{2} sin (2 x) + \frac{5}{2} x sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{11}{4} sin (2 x) + \frac{5}{4} cos (2 x)

= \frac{1}{2} x^{2} sin (2 x) + \frac{5}{2} x sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{11}{4} sin (2 x) + \frac{5}{4} cos (2 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} sin (2 x) + \frac{5}{2} x sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{11}{4} sin (2 x) + \frac{5}{4} cos (2 x) + C

a re a y = x, g (y, x) = x^{2}, [0, 1]

\frac{d}{d x} (cos (sin^{3} (5 x - 5)))

y^{^{'}} = \frac{( y ^{3} + 1 )}{( 3 x y ^{2} )}

x \to + 1 lim (\frac{1 - ∣ x ∣}{x - 1})

\int \frac{x ^{3}}{1 + x ^{4}} d x