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Beliebt Rechnen >

integral from 0 to pi/6 of tan(2θ)

Lösung

\int_{0}^{\frac{π}{6}} tan (2 θ) d θ

Lösung

\frac{1}{2} ln (2)

+1

Dezimale

0.34657 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{6}} tan (2 θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \int_{1}^{\frac{1}{2}} - \frac{1}{2 u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{2}}^{1} - \frac{1}{2 u} d u

Entferne die Konstante:

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

= - (- \frac{1}{2} \cdot \int_{\frac{1}{2}}^{1} \frac{1}{u} d u)

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- \frac{1}{2} [ln ∣ u ∣]_{\frac{1}{2}}^{1})

Vereinfache

= \frac{1}{2} [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Berechne die Grenzen:

ln (2)

= \frac{1}{2} ln (2)

Graph

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