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

Lösung

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

\frac{3}{2} x + \frac{1}{4} sin (2 x) - 2 sin (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int 2 cos (x) d x = 2 sin (x)

\int 1 d x = x

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) - 2 sin (x) + x

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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

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