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

Lösung

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

5 + \frac{- 2 - ln ( 1 + 2 ) + ln ( 2 + 5 )}{2}

Dezimale

1.81009 \dots

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

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

\int_{\frac{π}{4}}^{a r c t a n (2)} sec (u) d u = ln (2 + 5) - ln (1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

Vereinfache

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2)) : [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

= [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (2)} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

Berechne die Grenzen:

5 - \frac{1}{2}

= 5 - \frac{1}{2} + \frac{1}{2} (ln (2 + 5) - ln (1 + 2))

Vereinfache

= 5 + \frac{- 2 - ln ( 1 + 2 ) + ln ( 2 + 5 )}{2}

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

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\int_{- 1}^{2} (5 - x^{2}) d x