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Beliebt Rechnen >

integral of sin^3(x)*sqrt(cos(x))

Lösung

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

Lösung

- \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

u (1 - u^{2}) : u - u^{\frac{5}{2}}

= - \int u - u^{\frac{5}{2}} 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

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= - (\frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}})

Setze in

u = cos (x)

ein

= - (\frac{2}{3} cos^{\frac{3}{2}} (x) - \frac{2}{7} cos^{\frac{7}{2}} (x))

Vereinfache

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

Graph

Beliebte Beispiele

integral of (2x^3+2x^2+1)/(x^2-x)

\int \frac{2 x ^{3} + 2 x ^{2} + 1}{x ^{2} - x} d x

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\int 10 x - 4 x^{3} d x

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\int k x k x^{2} + 10 d x

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