integral from 0 to a of sin^2(x)

Lösung

\int_{0}^{a} sin^{2} (x) d x

\frac{1}{2} (a - \frac{1}{2} sin (2 a))

Schritte zur Lösung

\int_{0}^{a} sin^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{a} \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}^{a} 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}^{a} 1 d x - \int_{0}^{a} cos (2 x) d x)

\int_{0}^{a} 1 d x = a

\int_{0}^{a} cos (2 x) d x = \frac{1}{2} sin (2 a)

= \frac{1}{2} (a - \frac{1}{2} sin (2 a))

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

\int_{0}^{1} \frac{1}{( 1 + e ^{x} ) ^{2}} d x

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

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

\int_{- 2}^{0} 2 e^{4 x - 1} d x