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

Lösung

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

\frac{π}{4} + \frac{1}{2}

Dezimale

1.28539 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \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

= 2 \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

= 2 \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}

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

Vereinfache

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

= \frac{π}{4} + \frac{1}{2}

\int_{- π}^{0} x cos ((2 n + 1) x) d x

\int_{7}^{2} x d x

\int_{- 2}^{2} - 2 - x^{2} + 4 d x

\int_{1}^{5} (\frac{ln ( y )}{x y}) d y

\int_{x^{3}}^{x^{5}} (4 t + 4)^{6} d t