integral of (x-1)^3sin(x)

Lösung

\int (x - 1)^{3} sin (x) d x

- x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} cos (x) + 3 x cos (x) - 6 x sin (x) - 3 sin (x) - 5 cos (x) + C

Schritte zur Lösung

\int (x - 1)^{3} sin (x) d x

Multipliziere aus

(x - 1)^{3} sin (x) : x^{3} sin (x) - 3 x^{2} sin (x) + 3 x sin (x) - sin (x)

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

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

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

\int 3 x sin (x) d x = 3 (- x cos (x) + sin (x))

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

= - x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 3 (- x cos (x) + sin (x)) - (- cos (x))

Vereinfache

- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 3 (- x cos (x) + sin (x)) - (- cos (x)) : - x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} cos (x) + 3 x cos (x) - 6 x sin (x) - 3 sin (x) - 5 cos (x)

= - x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} cos (x) + 3 x cos (x) - 6 x sin (x) - 3 sin (x) - 5 cos (x)

Füge eine Konstante zur Lösung hinzu

= - x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} cos (x) + 3 x cos (x) - 6 x sin (x) - 3 sin (x) - 5 cos (x) + C

\int \frac{1}{1 + y ^{4}} d y

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

\int \frac{3 z - 6}{3} d z

\int sin (2 θ) cos (2 θ) d θ

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