integral from 2 to 4 of sqrt(3+x^2)

Lösung

\int_{2}^{4} 3 + x^{2} d x

219 - 7 + \frac{3}{2} ln (\frac{4}{3} + \frac{19}{3}) - \frac{3}{2} ln (\frac{2}{3} + \frac{7}{3})

Dezimale

6.95310 \dots

Schritte zur Lösung

\int_{2}^{4} 3 + x^{2} d x

Trigonometrische Substitution anwenden

= \int_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} 3 sec^{3} (u) d u

Entferne die Konstante:

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

= 3 \cdot \int_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} sec^{3} (u) d u

Wende integrale Reduktion an

= 3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} + \frac{1}{2} \cdot \int_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} sec (u) d u

\int_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} sec (u) d u = ln (\frac{4}{3} + \frac{19}{3}) - ln (\frac{2}{3} + \frac{7}{3})

= 3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} + \frac{1}{2} (ln (\frac{4}{3} + \frac{19}{3}) - ln (\frac{2}{3} + \frac{7}{3}))

Vereinfache

3 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} + \frac{1}{2} (ln (\frac{4}{3} + \frac{19}{3}) - ln (\frac{2}{3} + \frac{7}{3})) : 3 [\frac{1}{2} sec (u) tan (u)]_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} + \frac{3}{2} ln (\frac{4}{3} + \frac{19}{3}) - \frac{3}{2} ln (\frac{2}{3} + \frac{7}{3})

= 3 [\frac{1}{2} sec (u) tan (u)]_{a r c t a n (\frac{2}{3})}^{a r c t a n (\frac{4}{3})} + \frac{3}{2} ln (\frac{4}{3} + \frac{19}{3}) - \frac{3}{2} ln (\frac{2}{3} + \frac{7}{3})

Berechne die Grenzen:

\frac{2 19 - 7}{3}

= 3 \cdot \frac{2 19 - 7}{3} + \frac{3}{2} ln (\frac{4}{3} + \frac{19}{3}) - \frac{3}{2} ln (\frac{2}{3} + \frac{7}{3})

Vereinfache

= 219 - 7 + \frac{3}{2} ln (\frac{4}{3} + \frac{19}{3}) - \frac{3}{2} ln (\frac{2}{3} + \frac{7}{3})

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