integral from 0 to pi/2 of tan(x/2)

Lösung

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

ln (2)

Dezimale

0.69314 \dots

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

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

Vereinfache

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

= 2 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Berechne die Grenzen:

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

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

Vereinfache

= ln (2)

25 y^{^{''}} - 40 y^{^{'}} + 16 y = 0, y (0) = 4, y^{^{'}} (0) = - 5

\int ln (x^{\frac{1}{2}}) d x

\frac{\partial}{\partial x} (e^{e^{x}})

\frac{d}{d θ} (sin (θ) + cos (θ))

\int (x + 4) cos (nπ x) d x