integral of (x-1)^2(sin(x))^2

Lösung

\int (x - 1)^{2} (sin (x))^{2} d x

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

Schritte zur Lösung

\int (x - 1)^{2} (sin (x))^{2} d x

Vereinfache

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

\frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - \frac{1}{4} x cos (2 x) + \frac{1}{8} sin (2 x) - (x^{2} - \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x)

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (t) = 1.3 e^{0.7 t}

\int x \cdot e^{1 - x} d x

s im pl i f y (2 x - 5)^{4} (8 x^{2} - 5)^{- 3}

\frac{\partial}{\partial y} (y e^{- x y})

\frac{d}{d x} (ln (\frac{e ^{x}}{1 + e ^{x}}))