integral of sin^2(θ)cos^5(θ)

Lösung

\int sin^{2} (θ) cos^{5} (θ) d θ

\frac{sin ^{3} ( θ )}{3} - \frac{2 sin ^{5} ( θ )}{5} + \frac{sin ^{7} ( θ )}{7} + C

Schritte zur Lösung

\int sin^{2} (θ) cos^{5} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (θ))^{2} cos (θ) sin^{2} (θ) d θ

Wende U-Substitution an

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

Multipliziere aus

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

= \int u^{2} - 2 u^{4} + u^{6} 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^{2} d u - \int 2 u^{4} d u + \int u^{6} d u

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

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

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

= \frac{u ^{3}}{3} - \frac{2 u ^{5}}{5} + \frac{u ^{7}}{7}

Setze in

u = sin (θ)

ein

= \frac{sin ^{3} ( θ )}{3} - \frac{2 sin ^{5} ( θ )}{5} + \frac{sin ^{7} ( θ )}{7}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{3} ( θ )}{3} - \frac{2 sin ^{5} ( θ )}{5} + \frac{sin ^{7} ( θ )}{7} + C

\frac{\partial}{\partial x} (e^{5 x y})

\int \frac{x}{3 x} d x

\frac{d}{d x} (sec^{2} (x) tan^{2} (x))

\int \frac{2 x + 4}{x ^{2} - x} d x

\frac{\partial}{\partial x} (\frac{1}{3} x^{\frac{3}{2}})