integral from 1 to 3 of sqrt(x^2+8)

Lösung

\int_{1}^{3} x^{2} + 8 d x

\frac{3 17 - 3}{2} + ln (\frac{( 3 2 + 34 ) ^{4}}{1024})

Dezimale

6.99285 \dots

Schritte zur Lösung

\int_{1}^{3} x^{2} + 8 d x

Trigonometrische Substitution anwenden

= \int_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} 8 sec^{3} (u) d u

Entferne die Konstante:

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

= 8 \cdot \int_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} sec^{3} (u) d u

Wende integrale Reduktion an

= 8 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} + \frac{1}{2} \cdot \int_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} sec (u) d u

\int_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} sec (u) d u = ln (\frac{3 2 + 34}{4}) - \frac{1}{2} ln (2)

= 8 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} + \frac{1}{2} (ln (\frac{3 2 + 34}{4}) - \frac{1}{2} ln (2))

Vereinfache

8 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (\frac{1}{2 2})}^{a r c t a n (\frac{3}{2 2})} + \frac{1}{2} (ln (\frac{3 2 + 34}{4}) - \frac{1}{2} ln (2)) : 8 [\frac{1}{2} sec (u) tan (u)]_{a r c t a n (\frac{2}{4})}^{a r c t a n (\frac{3 2}{4})} + 4 ln (\frac{3 2 + 34}{4}) - 2 ln (2)

= 8 [\frac{1}{2} sec (u) tan (u)]_{a r c t a n (\frac{2}{4})}^{a r c t a n (\frac{3 2}{4})} + 4 ln (\frac{3 2 + 34}{4}) - 2 ln (2)

Berechne die Grenzen:

\frac{3 17}{16} - \frac{3}{16}

= 8 (\frac{3 17}{16} - \frac{3}{16}) + 4 ln (\frac{3 2 + 34}{4}) - 2 ln (2)

Vereinfache

= \frac{3 17 - 3}{2} + ln (\frac{( 3 2 + 34 ) ^{4}}{1024})

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