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integral of sin^7(3/5 x)

Lösung

\int sin^{7} (\frac{3}{5} x) d x

- \frac{5}{3} cos (\frac{3}{5} x) + \frac{5}{3} cos^{3} (\frac{3}{5} x) - cos^{5} (\frac{3}{5} x) + \frac{5}{21} cos^{7} (\frac{3 x}{5}) + C

Schritte zur Lösung

\int sin^{7} (\frac{3}{5} x) d x

Wende U-Substitution an

= \int sin^{7} (u) \frac{5}{3} d u

Entferne die Konstante:

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

= \frac{5}{3} \cdot \int sin^{7} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{5}{3} \cdot \int (1 - cos^{2} (u))^{3} sin (u) d u

Wende U-Substitution an

= \frac{5}{3} \cdot \int - (1 - v^{2})^{3} d v

Multipliziere aus

- (1 - v^{2})^{3} : - 1 + 3 v^{2} - 3 v^{4} + v^{6}

= \frac{5}{3} \cdot \int - 1 + 3 v^{2} - 3 v^{4} + v^{6} 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

= \frac{5}{3} (- \int 1 d v + \int 3 v^{2} d v - \int 3 v^{4} d v + \int v^{6} d v)

\int 1 d v = v

\int 3 v^{2} d v = v^{3}

\int 3 v^{4} d v = \frac{3 v ^{5}}{5}

\int v^{6} d v = \frac{v ^{7}}{7}

= \frac{5}{3} (- v + v^{3} - \frac{3 v ^{5}}{5} + \frac{v ^{7}}{7})

Ersetze zurück

= \frac{5}{3} (- cos (\frac{3}{5} x) + cos^{3} (\frac{3}{5} x) - \frac{3 cos ^{5} ( \frac{3}{5} x )}{5} + \frac{cos ^{7} ( \frac{3}{5} x )}{7})

Vereinfache

\frac{5}{3} (- cos (\frac{3}{5} x) + cos^{3} (\frac{3}{5} x) - \frac{3 cos ^{5} ( \frac{3}{5} x )}{5} + \frac{cos ^{7} ( \frac{3}{5} x )}{7}) : - \frac{5}{3} cos (\frac{3}{5} x) + \frac{5}{3} cos^{3} (\frac{3}{5} x) - cos^{5} (\frac{3}{5} x) + \frac{5}{21} cos^{7} (\frac{3 x}{5})

= - \frac{5}{3} cos (\frac{3}{5} x) + \frac{5}{3} cos^{3} (\frac{3}{5} x) - cos^{5} (\frac{3}{5} x) + \frac{5}{21} cos^{7} (\frac{3 x}{5})

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{3} cos (\frac{3}{5} x) + \frac{5}{3} cos^{3} (\frac{3}{5} x) - cos^{5} (\frac{3}{5} x) + \frac{5}{21} cos^{7} (\frac{3 x}{5}) + C

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