integral from 0 to 3 of 3sqrt(1+36x^2)

Lösung

\int_{0}^{3} 3 1 + 36 x^{2} d x

\frac{1}{2} (4513 + \frac{1}{2} ln (18 + 513))

Dezimale

82.02097 \dots

Schritte zur Lösung

\int_{0}^{3} 3 1 + 36 x^{2} d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{3} 1 + 36 x^{2} d x

Trigonometrische Substitution anwenden

= 3 \cdot \int_{0}^{a r c t a n (18)} \frac{1}{6} sec^{3} (u) d u

Entferne die Konstante:

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

= 3 \cdot \frac{1}{6} \cdot \int_{0}^{a r c t a n (18)} sec^{3} (u) d u

Wende integrale Reduktion an

= 3 \cdot \frac{1}{6} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (18)} + \frac{1}{2} \cdot \int_{0}^{a r c t a n (18)} sec (u) d u)

\int_{0}^{a r c t a n (18)} sec (u) d u = ln (18 + 513)

= 3 \cdot \frac{1}{6} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (18)} + \frac{1}{2} ln (18 + 513))

Vereinfache

3 \cdot \frac{1}{6} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{a r c t a n (18)} + \frac{1}{2} ln (18 + 513)) : \frac{1}{2} ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (18)} + \frac{1}{2} ln (18 + 513))

= \frac{1}{2} ([\frac{1}{2} sec (u) tan (u)]_{0}^{a r c t a n (18)} + \frac{1}{2} ln (18 + 513))

Berechne die Grenzen:

4513

= \frac{1}{2} (4513 + \frac{1}{2} ln (18 + 513))

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