integral from 1 to x of tan(t)

Lösung

\int_{1}^{x} tan (t) d t

- ln ∣ cos (x) ∣ + ln (cos (1))

Schritte zur Lösung

\int_{1}^{x} tan (t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{1}^{x} \frac{sin ( t )}{cos ( t )} d t

Wende U-Substitution an

= \int_{c o s (1)}^{c o s (x)} - \frac{1}{u} d u

Entferne die Konstante:

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

= - \int_{c o s (1)}^{c o s (x)} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= - [ln ∣ u ∣]_{c o s (1)}^{c o s (x)}

Berechne die Grenzen:

ln ∣ cos (x) ∣ - ln (cos (1))

= - (ln ∣ cos (x) ∣ - ln (cos (1)))

Vereinfache

= - ln ∣ cos (x) ∣ + ln (cos (1))

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

\int_{0}^{\frac{π}{2}} sin^{8} (x) d x

\int_{2}^{5} (3 x + 1) d x

\int_{1}^{\infty} x ln (x) d x

\int_{- 1}^{1} \frac{1}{1 - ∣ x ∣} d x