integral of cos(θ)(1-cos(θ))^{99}sin(θ)

Lösung

\int cos (θ) (1 - cos (θ))^{99} sin (θ) d θ

- \frac{1}{101} (1 - cos (θ))^{101} + \frac{1}{100} (1 - cos (θ))^{100} + C

Schritte zur Lösung

\int cos (θ) (1 - cos (θ))^{99} sin (θ) d θ

Wende U-Substitution an

= \int - u (1 - u)^{99} d u

Entferne die Konstante:

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

= - \int u (1 - u)^{99} d u

Wende U-Substitution an

= - \int - (- v + 1) v^{99} d v

Entferne die Konstante:

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

= - (- \int (- v + 1) v^{99} d v)

Multipliziere aus

(- v + 1) v^{99} : - v^{100} + v^{99}

= - (- \int - v^{100} + v^{99} d v)

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- (- \int v^{100} d v + \int v^{99} d v))

\int v^{100} d v = \frac{v ^{101}}{101}

\int v^{99} d v = \frac{v ^{100}}{100}

= - (- (- \frac{v ^{101}}{101} + \frac{v ^{100}}{100}))

Ersetze zurück

= - (- (- \frac{( 1 - cos ( θ ) ) ^{101}}{101} + \frac{( 1 - cos ( θ ) ) ^{100}}{100}))

Vereinfache

- (- (- \frac{( 1 - cos ( θ ) ) ^{101}}{101} + \frac{( 1 - cos ( θ ) ) ^{100}}{100})) : - \frac{1}{101} (1 - cos (θ))^{101} + \frac{1}{100} (1 - cos (θ))^{100}

= - \frac{1}{101} (1 - cos (θ))^{101} + \frac{1}{100} (1 - cos (θ))^{100}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{101} (1 - cos (θ))^{101} + \frac{1}{100} (1 - cos (θ))^{100} + C

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