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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

cos (x) (2 + sin (x))^{2} : 4 cos (x) + 4 cos (x) sin (x) + sin^{2} (x) cos (x)

= \int 4 cos (x) + 4 cos (x) sin (x) + sin^{2} (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 4 cos (x) d x + \int 4 cos (x) sin (x) d x + \int sin^{2} (x) cos (x) d x

\int 4 cos (x) d x = 4 sin (x)

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{5 x - 6}{3 x - 1} d x

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

\int \frac{2 x + 5}{5 x + 9} d x

\int \frac{( ln ( x ) + 1 )}{x} d x

\int e^{x^{3}} \cdot 3 x^{2} d x