integral from-1 to 1 of sqrt(x^2+2)

Lösung

\int_{- 1}^{1} x^{2} + 2 d x

3 + ln (\frac{1}{2} + \frac{3}{2}) - ln (- \frac{1}{2} + \frac{3}{2})

Dezimale

3.04900 \dots

Schritte zur Lösung

\int_{- 1}^{1} x^{2} + 2 d x

Trigonometrische Substitution anwenden

= \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} 2 sec^{3} (u) d u

Entferne die Konstante:

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

= 2 \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec^{3} (u) d u

Wende integrale Reduktion an

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

\int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{1}{2})} sec (u) d u = ln (\frac{1}{2} + \frac{3}{2}) - ln (- \frac{1}{2} + \frac{3}{2})

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{3}{2}

= 2 \cdot \frac{3}{2} + ln (\frac{1}{2} + \frac{3}{2}) - ln (- \frac{1}{2} + \frac{3}{2})

Vereinfache

= 3 + ln (\frac{1}{2} + \frac{3}{2}) - ln (- \frac{1}{2} + \frac{3}{2})

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