integral from 0 to pi/6 of (cos(x))^2

Lösung

\int_{0}^{\frac{π}{6}} (cos (x))^{2} d x

\frac{2 π + 3 3}{24}

Dezimale

0.47830 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{6}} (cos (x))^{2} d x

Vereinfache

= \int_{0}^{\frac{π}{6}} cos^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{\frac{π}{6}} \frac{1 + cos ( 2 x )}{2} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 1 + cos (2 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

= \frac{1}{2} (\int_{0}^{\frac{π}{6}} 1 d x + \int_{0}^{\frac{π}{6}} cos (2 x) d x)

\int_{0}^{\frac{π}{6}} 1 d x = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 x) d x = \frac{3}{4}

= \frac{1}{2} (\frac{π}{6} + \frac{3}{4})

Vereinfache

\frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{2 π + 3 3}{24}

= \frac{2 π + 3 3}{24}

\int_{2}^{4} 4 x - x^{2} d x

\int_{0}^{1} (4 x - 3)^{2} d x

\int_{0}^{2} \frac{3}{2} x^{2} d x

\int_{0}^{3} (3 t - 1)^{56} d t

\int_{- 2}^{1} 3 t^{2} - 1 d t