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Beliebt Rechnen >

integral of cos^6((5x)/2)

Lösung

\int cos^{6} (\frac{5 x}{2}) d x

Lösung

\frac{1}{240} (16 cos^{5} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 5 (4 cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 3 (5 x + sin (5 x)))) + C

Schritte zur Lösung

\int cos^{6} (\frac{5 x}{2}) d x

Wende U-Substitution an

= \int cos^{6} (u) \frac{2}{5} d u

Entferne die Konstante:

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

= \frac{2}{5} \cdot \int cos^{6} (u) d u

Wende integrale Reduktion an

= \frac{2}{5} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} \cdot \int cos^{4} (u) d u)

\int cos^{4} (u) d u = \frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))

= \frac{2}{5} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))

Setze in

u = \frac{5 x}{2}

ein

= \frac{2}{5} (\frac{sin ( \frac{5 x}{2} ) cos ^{5} ( \frac{5 x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + \frac{3}{8} (\frac{5 x}{2} + \frac{1}{2} sin (2 \cdot \frac{5 x}{2}))))

Vereinfache

\frac{2}{5} (\frac{sin ( \frac{5 x}{2} ) cos ^{5} ( \frac{5 x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + \frac{3}{8} (\frac{5 x}{2} + \frac{1}{2} sin (2 \cdot \frac{5 x}{2})))) : \frac{1}{240} (16 cos^{5} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 5 (4 cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 3 (5 x + sin (5 x))))

= \frac{1}{240} (16 cos^{5} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 5 (4 cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 3 (5 x + sin (5 x))))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{240} (16 cos^{5} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 5 (4 cos^{3} (\frac{5 x}{2}) sin (\frac{5 x}{2}) + 3 (5 x + sin (5 x)))) + C

Graph

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