integral from-pi/4 to pi/4 of sin^2(x)

Lösung

\int_{- \frac{π}{4}}^{\frac{π}{4}} sin^{2} (x) d x

\frac{π - 2}{4}

Dezimale

0.28539 \dots

Schritte zur Lösung

\int_{- \frac{π}{4}}^{\frac{π}{4}} sin^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{- \frac{π}{4}}^{\frac{π}{4}} \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_{- \frac{π}{4}}^{\frac{π}{4}} 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_{- \frac{π}{4}}^{\frac{π}{4}} 1 d x - \int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 x) d x)

\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d x = \frac{π}{2}

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 x) d x = 1

= \frac{1}{2} (\frac{π}{2} - 1)

Vereinfache

\frac{1}{2} (\frac{π}{2} - 1) : \frac{π - 2}{4}

= \frac{π - 2}{4}

\int_{1}^{5} \frac{2}{x ^{2}} d x

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

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

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

\int_{2}^{5} π (x^{2} - \frac{1}{5} x^{4}) d x