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

Lösung

\int_{2}^{3} 3 t^{2} + 1 d t

\frac{1}{3} (321 - 39 + \frac{1}{2} (ln (33 + 27) - ln (23 + 13)))

Dezimale

4.44555 \dots

Schritte zur Lösung

\int_{2}^{3} 3 t^{2} + 1 d t

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

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

\int_{a r c t a n (3 \cdot 2)}^{a r c t a n (3 \cdot 3)} sec (u) d u = ln (33 + 27) - ln (23 + 13)

= \frac{1}{3} [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (3 \cdot 2)}^{a r c t a n (3 \cdot 3)} + \frac{1}{2} (ln (33 + 27) - ln (23 + 13))

Vereinfache

\frac{1}{3} [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (3 \cdot 2)}^{a r c t a n (3 \cdot 3)} + \frac{1}{2} (ln (33 + 27) - ln (23 + 13)) : \frac{1}{3} ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (23)}^{a r c t a n (33)} + \frac{1}{2} (ln (33 + 27) - ln (23 + 13)))

= \frac{1}{3} ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (23)}^{a r c t a n (33)} + \frac{1}{2} (ln (33 + 27) - ln (23 + 13)))

Berechne die Grenzen:

321 - 39

= \frac{1}{3} (321 - 39 + \frac{1}{2} (ln (33 + 27) - ln (23 + 13)))

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