integral from 2 to 7 of 5arccot(x)

Lösung

\int_{2}^{7} 5 arccot (x) d x

5 (7 arccot (7) - 2 arccot (2) + \frac{ln ( 10 )}{2})

Dezimale

6.08638 \dots

Schritte zur Lösung

\int_{2}^{7} 5 arccot (x) d x

Entferne die Konstante:

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

= 5 \cdot \int_{2}^{7} arccot (x) d x

Wende die partielle Integration an

= 5 [x arccot (x) - \int - \frac{x}{x ^{2} + 1} d x]_{2}^{7}

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

= 5 [x arccot (x) - (- \frac{1}{2} ln x^{2} + 1)]_{2}^{7}

Vereinfache

= 5 [x arccot (x) + \frac{1}{2} ln x^{2} + 1]_{2}^{7}

Berechne die Grenzen:

7 arccot (7) - 2 arccot (2) + \frac{ln ( 50 ) - ln ( 5 )}{2}

= 5 (7 arccot (7) - 2 arccot (2) + \frac{ln ( 50 ) - ln ( 5 )}{2})

Vereinfache

= 5 (7 arccot (7) - 2 arccot (2) + \frac{ln ( 10 )}{2})

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