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Beliebt Rechnen >

integral from 0 to 5 of sqrt(x^2+25)

Lösung

\int_{0}^{5} x^{2} + 25 d x

Lösung

\frac{25 ( 2 + ln ( 1 + 2 ) )}{2}

+1

Dezimale

28.69483 \dots

Schritte zur Lösung

\int_{0}^{5} x^{2} + 25 d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{4}} 25 sec^{3} (u) d u

Entferne die Konstante:

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

= 25 \cdot \int_{0}^{\frac{π}{4}} sec^{3} (u) d u

Wende integrale Reduktion an

= 25 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} sec (u) d u)

\int_{0}^{\frac{π}{4}} sec (u) d u = ln (1 + 2)

= 25 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2))

Vereinfache

25 ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)) : 25 ([\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2))

= 25 ([\frac{1}{2} sec (u) tan (u)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2))

Berechne die Grenzen:

\frac{1}{2}

= 25 (\frac{1}{2} + \frac{1}{2} ln (1 + 2))

Vereinfache

= \frac{25 ( 2 + ln ( 1 + 2 ) )}{2}

Graph

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