integral from pi/3 to pi/6 of tan^2(x)

Lösung

\int_{\frac{π}{3}}^{\frac{π}{6}} tan^{2} (x) d x

\frac{π}{6} + \frac{1}{3} - 3

Dezimale

- 0.63110 \dots

Schritte zur Lösung

\int_{\frac{π}{3}}^{\frac{π}{6}} tan^{2} (x) d x

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

= - \int_{\frac{π}{6}}^{\frac{π}{3}} tan^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - \int_{\frac{π}{6}}^{\frac{π}{3}} - 1 + sec^{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

= - (- \int_{\frac{π}{6}}^{\frac{π}{3}} 1 d x + \int_{\frac{π}{6}}^{\frac{π}{3}} sec^{2} (x) d x)

\int_{\frac{π}{6}}^{\frac{π}{3}} 1 d x = \frac{π}{6}

\int_{\frac{π}{6}}^{\frac{π}{3}} sec^{2} (x) d x = 3 - \frac{1}{3}

= - (- \frac{π}{6} + 3 - \frac{1}{3})

Vereinfache

= \frac{π}{6} + \frac{1}{3} - 3

\int_{- 1}^{3} e^{x} - 3 d x

\int_{0}^{1} (1 - e^{2})^{\frac{1}{2}} d x

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

\int_{- π}^{π} (sin (x))^{2} d x

\int_{0}^{1} 1 + 2 t^{2} + t^{4} d t