integral of sin^3(x)-sin^5(x)

Lösung

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

\frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

Schritte zur Lösung

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

Wende die Summenregel an:

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

= \int sin^{3} (x) d x - \int sin^{5} (x) d x

\int sin^{3} (x) d x = - cos (x) + \frac{cos ^{3} ( x )}{3}

\int sin^{5} (x) d x = - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}

= - cos (x) + \frac{cos ^{3} ( x )}{3} - (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5})

Vereinfache

- cos (x) + \frac{cos ^{3} ( x )}{3} - (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}) : \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

d er i v a t i v e f (x) = \frac{x}{x ^{2} - 4}

9 x^{2} y^{^{'}} = y^{^{'}} + 6 x e^{- y}

y^{^{'}} = y (x y^{5} + 1)

a re a y = ∣ x ∣, y = x^{2} - 1

t a y l or (cos (x))^{\frac{1}{x}}