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

Lösung

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

\frac{3 ( π + 2 )}{8}

Dezimale

1.92809 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 3 \cdot \int_{0}^{\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

= 3 \cdot \frac{1}{2} \cdot \int_{0}^{\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

= 3 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{4}} 1 d x + \int_{0}^{\frac{π}{4}} cos (2 x) d x)

\int_{0}^{\frac{π}{4}} 1 d x = \frac{π}{4}

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

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

Vereinfache

3 \cdot \frac{1}{2} (\frac{π}{4} + \frac{1}{2}) : \frac{3 ( π + 2 )}{8}

= \frac{3 ( π + 2 )}{8}

\int_{0}^{5} \frac{x + 5}{x ^{2} + x - 20} d x

\int_{- 3}^{0} 1 - x + 1 + x d x

\int_{3}^{\infty} \frac{1}{x ^{2} + 9} d x

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

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