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

Lösung

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

- x^{2} cos (x) + 2 x sin (x) - 6 x cos (x) - 7 cos (x) + 6 sin (x) + C

Schritte zur Lösung

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

x + 3 = 3 (\frac{x}{3} + 1)

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

Vereinfache

(3 (\frac{x}{3} + 1))^{2} sin (x) : x^{2} sin (x) + 6 x sin (x) + 9 sin (x)

= \int x^{2} sin (x) + 6 x sin (x) + 9 sin (x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} sin (x) d x + \int 6 x sin (x) d x + \int 9 sin (x) d x

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

\int 6 x sin (x) d x = 6 (- x cos (x) + sin (x))

\int 9 sin (x) d x = - 9 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 6 (- x cos (x) + sin (x)) - 9 cos (x)

Vereinfache

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 6 (- x cos (x) + sin (x)) - 9 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 6 x cos (x) - 7 cos (x) + 6 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 6 x cos (x) - 7 cos (x) + 6 sin (x)

Füge eine Konstante zur Lösung hinzu

= - x^{2} cos (x) + 2 x sin (x) - 6 x cos (x) - 7 cos (x) + 6 sin (x) + C

\frac{d}{d x} (sin (- 4 x))

(x^{2} + 1) \frac{d y}{d x} + 4 x y = x

d er i v a t i v e \frac{( x )}{x + 1}

\int \frac{x}{x ^{4} + 8 x ^{2} + 17} d x

\frac{\partial}{\partial x} (y cos (x))