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Beliebt Rechnen >

integral of (2cos(4x)-1)^{1/2}sin(4x)

Lösung

\int (2 cos (4 x) - 1)^{\frac{1}{2}} sin (4 x) d x

Lösung

- \frac{1}{12} (2 cos (4 x) - 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (2 cos (4 x) - 1)^{\frac{1}{2}} sin (4 x) d x

Wende U-Substitution an

= \int - \frac{u ^{\frac{1}{2}}}{8} d u

Entferne die Konstante:

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

= - \frac{1}{8} \cdot \int u^{\frac{1}{2}} d u

Wende die Potenzregel an

= - \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = 2 cos (4 x) - 1

ein

= - \frac{1}{8} \cdot \frac{2}{3} (2 cos (4 x) - 1)^{\frac{3}{2}}

Vereinfache

- \frac{1}{8} \cdot \frac{2}{3} (2 cos (4 x) - 1)^{\frac{3}{2}} : - \frac{1}{12} (2 cos (4 x) - 1)^{\frac{3}{2}}

= - \frac{1}{12} (2 cos (4 x) - 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{12} (2 cos (4 x) - 1)^{\frac{3}{2}} + C

Graph

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