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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (2 + cos (u))^{2} d u

Multipliziere aus

(2 + cos (u))^{2} : 4 + 4 cos (u) + cos^{2} (u)

= \frac{1}{2} \cdot \int 4 + 4 cos (u) + cos^{2} (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 4 d u + \int 4 cos (u) d u + \int cos^{2} (u) d u)

\int 4 d u = 4 u

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

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

= \frac{1}{2} (4 u + 4 sin (u) + \frac{1}{2} (u + \frac{1}{2} sin (2 u)))

Setze in

u = 2 x

ein

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

Vereinfache

\frac{1}{2} (4 \cdot 2 x + 4 sin (2 x) + \frac{1}{2} (2 x + \frac{1}{2} sin (2 \cdot 2 x))) : \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x))

= \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x)) + C

Graph

Beliebte Beispiele

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\int \frac{4}{x ^{5}} d x