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

Lösung

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

\frac{8}{3}

Dezimale

2.66666 \dots

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{0}^{\frac{π}{4}} sec^{4} (u) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{\frac{π}{4}} sec^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int_{0}^{\frac{π}{4}} (1 + tan^{2} (u)) sec^{2} (u) d u

Wende U-Substitution an

= 2 \cdot \int_{0}^{1} 1 + v^{2} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int_{0}^{1} 1 d v + \int_{0}^{1} v^{2} d v)

\int_{0}^{1} 1 d v = 1

\int_{0}^{1} v^{2} d v = \frac{1}{3}

= 2 (1 + \frac{1}{3})

Vereinfache

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

= \frac{8}{3}

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