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Beliebt Rechnen >

integral of [cos(4x)+cos(8x)]^2

Lösung

\int [cos (4 x) + cos (8 x)]^{2} d x

Lösung

\frac{1}{96} (96 x + 64 sin^{3} (4 x) + 6 sin (8 x) + 48 cos (8 x) sin (4 x) + 3 sin (16 x)) + C

Schritte zur Lösung

\int [cos (4 x) + cos (8 x)]^{2} d x

Vereinfache

(cos (4 x) + cos (8 x))^{2} : (cos (4 x) + cos (8 x)) (cos (4 x) + cos (8 x))

= \int (cos (4 x) + cos (8 x)) (cos (4 x) + cos (8 x)) d x

Vereinfache

(cos (4 x) + cos (8 x)) (cos (4 x) + cos (8 x)) : (cos (4 x) + cos (8 x))^{2}

= \int (cos (4 x) + cos (8 x))^{2} d x

Multipliziere aus

(cos (4 x) + cos (8 x))^{2} : cos^{2} (4 x) + 2 cos (4 x) cos (8 x) + cos^{2} (8 x)

= \int cos^{2} (4 x) + 2 cos (4 x) cos (8 x) + cos^{2} (8 x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

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

\int 2 cos (u) cos (2 u) d u = 2 (cos (2 u) sin (u) + \frac{4}{3} sin^{3} (u))

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

= \frac{1}{4} (\frac{1}{2} (u + \frac{1}{2} sin (2 u)) + 2 (cos (2 u) sin (u) + \frac{4}{3} sin^{3} (u)) + \frac{1}{2} (u + \frac{1}{4} sin (4 u)))

Setze in

u = 4 x

ein

= \frac{1}{4} (\frac{1}{2} (4 x + \frac{1}{2} sin (2 \cdot 4 x)) + 2 (cos (2 \cdot 4 x) sin (4 x) + \frac{4}{3} sin^{3} (4 x)) + \frac{1}{2} (4 x + \frac{1}{4} sin (4 \cdot 4 x)))

Vereinfache

\frac{1}{4} (\frac{1}{2} (4 x + \frac{1}{2} sin (2 \cdot 4 x)) + 2 (cos (2 \cdot 4 x) sin (4 x) + \frac{4}{3} sin^{3} (4 x)) + \frac{1}{2} (4 x + \frac{1}{4} sin (4 \cdot 4 x))) : \frac{1}{96} (96 x + 64 sin^{3} (4 x) + 6 sin (8 x) + 48 cos (8 x) sin (4 x) + 3 sin (16 x))

= \frac{1}{96} (96 x + 64 sin^{3} (4 x) + 6 sin (8 x) + 48 cos (8 x) sin (4 x) + 3 sin (16 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{96} (96 x + 64 sin^{3} (4 x) + 6 sin (8 x) + 48 cos (8 x) sin (4 x) + 3 sin (16 x)) + C

Graph

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