integral from 0 to sqrt(3 of)sqrt(1+x^2)

Lösung

\int_{0}^{3} 1 + x^{2} d x

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

Dezimale

2.39052 \dots

Schritte zur Lösung

\int_{0}^{3} 1 + x^{2} d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{3}} sec^{3} (u) d u

Wende integrale Reduktion an

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{3}} sec (u) d u

\int_{0}^{\frac{π}{3}} sec (u) d u = ln (2 + 3)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

Vereinfache

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3) : [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

= [\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)

Berechne die Grenzen:

3

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

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