integral from 0 to pi/2 of 1/(cos(x)+2)

Lösung

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

\frac{π}{3 3}

Dezimale

0.60459 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende integrale Substitution an

= 2 \cdot \int_{0}^{\frac{1}{3}} \frac{1}{3 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 2 \cdot \frac{1}{3} [arctan (v)]_{0}^{\frac{1}{3}}

Vereinfache

2 \cdot \frac{1}{3} [arctan (v)]_{0}^{\frac{1}{3}} : \frac{2}{3} [arctan (v)]_{0}^{\frac{1}{3}}

= \frac{2}{3} [arctan (v)]_{0}^{\frac{1}{3}}

Berechne die Grenzen:

\frac{π}{6}

= \frac{2}{3} \cdot \frac{π}{6}

Vereinfache

= \frac{π}{3 3}

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