integral from 1 to 5 of (x-arctan(x))

Lösung

\int_{1}^{5} (x - arctan (x)) d x

12 - 5 arctan (5) + \frac{1}{2} ln (26) + \frac{π}{4} - \frac{1}{2} ln (2)

Dezimale

7.20086 \dots

Schritte zur Lösung

\int_{1}^{5} x - arctan (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_{1}^{5} x d x - \int_{1}^{5} arctan (x) d x

\int_{1}^{5} x d x = 12

\int_{1}^{5} arctan (x) d x = 5 arctan (5) - \frac{1}{2} ln (26) - \frac{π}{4} + \frac{1}{2} ln (2)

= 12 - (5 arctan (5) - \frac{1}{2} ln (26) - \frac{π}{4} + \frac{1}{2} ln (2))

Vereinfache

= 12 - 5 arctan (5) + \frac{1}{2} ln (26) + \frac{π}{4} - \frac{1}{2} ln (2)