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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin^{2} (x) + 2 x - sin (2 x) : \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x)

= \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x) + C

Graph

Beliebte Beispiele

integral of (x^2+3x+8)/(x^3-x^2+3x-3)

\int \frac{x ^{2} + 3 x + 8}{x ^{3} - x ^{2} + 3 x - 3} d x

integral of 16cos^4(5pi)t

\int 16 cos^{4} (5 π) t d t

integral of x/(3x+5)

\int \frac{x}{3 x + 5} d x

integral of x^3(x+7)^9

\int x^{3} (x + 7)^{9} d x

integral of ((2+x)^2)/(x^2)

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