integral from a to b of arctan(x)

Lösung

\int_{a}^{b} arctan (x) d x

b arctan (b) - \frac{1}{2} ln b^{2} + 1 - a arctan (a) + \frac{1}{2} ln a^{2} + 1

Schritte zur Lösung

\int_{a}^{b} arctan (x) d x

Wende die partielle Integration an

= [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{a}^{b}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

= [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{a}^{b}

Berechne die Grenzen:

b arctan (b) - \frac{1}{2} ln b^{2} + 1 - a arctan (a) + \frac{1}{2} ln a^{2} + 1

= b arctan (b) - \frac{1}{2} ln b^{2} + 1 - a arctan (a) + \frac{1}{2} ln a^{2} + 1

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