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Beliebt Rechnen >

integral from-2 to 2 of sqrt(1+100x^2)

Lösung

\int_{- 2}^{2} 1 + 100 x^{2} d x

Lösung

\frac{1}{10} (20401 + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401)))

+1

Dezimale

40.41891 \dots

Schritte zur Lösung

\int_{- 2}^{2} 1 + 100 x^{2} d x

Trigonometrische Substitution anwenden

= \int_{- a r c t a n (20)}^{a r c t a n (20)} \frac{1}{10} sec^{3} (u) d u

Entferne die Konstante:

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

= \frac{1}{10} \cdot \int_{- a r c t a n (20)}^{a r c t a n (20)} sec^{3} (u) d u

Wende integrale Reduktion an

= \frac{1}{10} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (20)}^{a r c t a n (20)} + \frac{1}{2} \cdot \int_{- a r c t a n (20)}^{a r c t a n (20)} sec (u) d u)

\int_{- a r c t a n (20)}^{a r c t a n (20)} sec (u) d u = ln (20 + 401) - ln (- 20 + 401)

= \frac{1}{10} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (20)}^{a r c t a n (20)} + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401)))

Vereinfache

\frac{1}{10} ([\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (20)}^{a r c t a n (20)} + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401))) : \frac{1}{10} ([\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (20)}^{a r c t a n (20)} + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401)))

= \frac{1}{10} ([\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (20)}^{a r c t a n (20)} + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401)))

Berechne die Grenzen:

20401

= \frac{1}{10} (20401 + \frac{1}{2} (ln (20 + 401) - ln (- 20 + 401)))

Graph

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