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

Lösung

\int (x^{2} + 2 x + 3) cos (2 x) d x

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

Schritte zur Lösung

\int (x^{2} + 2 x + 3) cos (2 x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int_{- \infty}^{0} 5^{x} d x

\int_{2}^{4} \frac{x ^{2} + 8}{x ^{2}} d x

\int (\frac{6 x ^{5} 12 x ^{4} + 9 x ^{3}}{3 x ^{2}}) d x

in v erse l a pl a ce \frac{1}{( s ^{2} + 1 ) ( s ^{2} + 4 s + 8 )}

d er i v a t i v e f (x) = x^{2} + 5