integral from 1 to N of x/(sqrt(1+x^2))

Lösung

\int_{1}^{N} \frac{x}{1 + x ^{2}} d x

N^{2} + 1 - 2

Schritte zur Lösung

\int_{1}^{N} \frac{x}{1 + x ^{2}} d x

Wende U-Substitution an

= \int_{2}^{1 + N^{2}} \frac{1}{2 u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{2}^{1 + N^{2}} \frac{1}{u} d u

Wende die Potenzregel an

= \frac{1}{2} [2 u^{\frac{1}{2}}]_{2}^{1 + N^{2}}

Wende Radikal Regel an:

a^{\frac{1}{n}} = n a

= \frac{1}{2} [2 u]_{2}^{1 + N^{2}}

Berechne die Grenzen:

2 1 + N^{2} - 22

= \frac{1}{2} (2 1 + N^{2} - 22)

Vereinfache

= N^{2} + 1 - 2

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