integral from 5 to 10 of sqrt(x^2+5)

Lösung

\int_{5}^{10} x^{2} + 5 d x

5 (105 - \frac{15}{2} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6)))

Dezimale

39.18869 \dots

Schritte zur Lösung

\int_{5}^{10} x^{2} + 5 d x

Trigonometrische Substitution anwenden

= \int_{a r c t a n (5)}^{a r c t a n (25)} 5 sec^{3} (u) d u

Entferne die Konstante:

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

= 5 \cdot \int_{a r c t a n (5)}^{a r c t a n (25)} sec^{3} (u) d u

Wende integrale Reduktion an

= 5 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (5)}^{a r c t a n (25)} + \frac{1}{2} \cdot \int_{a r c t a n (5)}^{a r c t a n (25)} sec (u) d u

\int_{a r c t a n (5)}^{a r c t a n (25)} sec (u) d u = ln (25 + 21) - ln (5 + 6)

= 5 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (5)}^{a r c t a n (25)} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6))

Vereinfache

5 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (5)}^{a r c t a n (25)} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6)) : 5 ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (5)}^{a r c t a n (25)} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6)))

= 5 ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (5)}^{a r c t a n (25)} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6)))

Berechne die Grenzen:

105 - \frac{15}{2}

= 5 (105 - \frac{15}{2} + \frac{1}{2} (ln (25 + 21) - ln (5 + 6)))

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