integral from 0 to pi/2 of-4sin^2(3x)

Lösung

\int_{0}^{\frac{π}{2}} - 4 sin^{2} (3 x) d x

- π

Dezimale

- 3.14159 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} - 4 sin^{2} (3 x) d x

Entferne die Konstante:

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

= - 4 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (3 x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - 4 \cdot \int_{0}^{\frac{π}{2}} \frac{1}{2} (1 - cos (6 x)) d x

Entferne die Konstante:

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

= - 4 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (6 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

= - 4 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d x - \int_{0}^{\frac{π}{2}} cos (6 x) d x)

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

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

= - 4 \cdot \frac{1}{2} (\frac{π}{2} - 0)

Vereinfache

- 4 \cdot \frac{1}{2} (\frac{π}{2} - 0) : - π

= - π

\int \frac{1}{x ^{2}} d x

x \to 0 lim (\frac{tan ^{2} ( x )}{1 - cos ( x )})

d er i v a t i v e \frac{x ^{2}}{x + 2}

\frac{\partial}{\partial x} (x y + \frac{1}{x} + \frac{1}{y})

t an g e n t f (x) = 9 - 6 x, at x = - 1