integral from 0 to 2pi of (cos(t))^2

Lösung

\int_{0}^{2 π} (cos (t))^{2} d t

π

Dezimale

3.14159 \dots

Schritte zur Lösung

\int_{0}^{2 π} (cos (t))^{2} d t

Vereinfache

= \int_{0}^{2 π} cos^{2} (t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

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

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

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}^{2 π} 1 d t + \int_{0}^{2 π} cos (2 t) d t)

\int_{0}^{2 π} 1 d t = 2 π

\int_{0}^{2 π} cos (2 t) d t = 0

= \frac{1}{2} (2 π + 0)

Vereinfache

\frac{1}{2} (2 π + 0) : π

= π

\int_{0}^{1} (5^{x} - 2^{x}) d x

\int_{0}^{3} 2 x (x - 6) d x

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

\int_{0}^{2} 2 x e^{x} - e^{x} + e^{- x} d x

\int_{0}^{4} x^{2} e^{x} d x