integral of (1+sqrt(cos(x)))^2sin(x)

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

l a pl a ce t r an s f or m t^{2} e^{- 2 t} sin (3 t + 30)

\frac{\partial}{\partial x} (y x^{2} + y^{2})

x \to \infty lim (3 x^{3} - 1000 x^{2})

\int \frac{1}{x ^{2} - 2 x - 3} d x

\int 121 - t^{2} d t