integral from-pi/2 to pi/2 of cos^3(x)

Lösung

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (x) d x

\frac{4}{3}

Dezimale

1.33333 \dots

Schritte zur Lösung

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{- \frac{π}{2}}^{\frac{π}{2}} (1 - sin^{2} (x)) cos (x) d x

Wende U-Substitution an

= \int_{- 1}^{1} 1 - u^{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_{- 1}^{1} 1 d u - \int_{- 1}^{1} u^{2} d u

\int_{- 1}^{1} 1 d u = 2

\int_{- 1}^{1} u^{2} d u = \frac{2}{3}

= 2 - \frac{2}{3}

Vereinfache

2 - \frac{2}{3} : \frac{4}{3}

= \frac{4}{3}

\int_{0}^{\frac{π}{2}} cos (θ) sin (θ) d θ

\int_{- 1}^{1} x cos (x) d x

\int_{0}^{π} x^{2} d x

\int_{0}^{π} cos^{2} (θ) sin (θ) d θ

\int_{0}^{π} sin (x) sin (x) d x