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Beliebt Rechnen >

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

Lösung

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

Lösung

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

+1

Dezimale

2.29558 \dots

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{4}}^{\frac{π}{4}} sec^{3} (u) d u

Wende integrale Reduktion an

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- \frac{π}{4}}^{\frac{π}{4}} + \frac{1}{2} \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} sec (u) d u

\int_{- \frac{π}{4}}^{\frac{π}{4}} sec (u) d u = ln (1 + 2) - ln (- 1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- \frac{π}{4}}^{\frac{π}{4}} + \frac{1}{2} (ln (1 + 2) - ln (- 1 + 2))

Vereinfache

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- \frac{π}{4}}^{\frac{π}{4}} + \frac{1}{2} (ln (1 + 2) - ln (- 1 + 2)) : [\frac{1}{2} sec (u) tan (u)]_{- \frac{π}{4}}^{\frac{π}{4}} + \frac{1}{2} (ln (1 + 2) - ln (- 1 + 2))

= [\frac{1}{2} sec (u) tan (u)]_{- \frac{π}{4}}^{\frac{π}{4}} + \frac{1}{2} (ln (1 + 2) - ln (- 1 + 2))

Berechne die Grenzen:

2

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

Graph

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