integral of cos^3(sqrt(x))

Lösung

\int cos^{3} (x) d x

2 (x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)) + C

Schritte zur Lösung

\int cos^{3} (x) d x

Wende U-Substitution an

= \int cos^{3} (u) \cdot 2 u d u

Entferne die Konstante:

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

= 2 \cdot \int cos^{3} (u) u d u

Wende die partielle Integration an

= 2 (u (sin (u) - \frac{1}{3} sin^{3} (u)) - \int sin (u) - \frac{1}{3} sin^{3} (u) d u)

\int sin (u) - \frac{1}{3} sin^{3} (u) d u = - \frac{2}{3} cos (u) - \frac{1}{9} cos^{3} (u)

= 2 (u (sin (u) - \frac{1}{3} sin^{3} (u)) - (- \frac{2}{3} cos (u) - \frac{1}{9} cos^{3} (u)))

Setze in

u = x

ein

= 2 (x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)))

Multipliziere aus

x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)) : x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

= 2 (x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x))

Füge eine Konstante zur Lösung hinzu

= 2 (x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)) + C

in t e g r a l \frac{ln ( x )}{x}

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