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

Lösung

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

- \frac{ln ( 5 + 1 ) - ln ( ( 5 - 1 ) ( 2 + 1 ) ) + ln ( 2 - 1 )}{2}

Dezimale

0.40016 \dots

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{2}^{5} \frac{1}{u ^{2} - 1} d u

Faktorisiere

u^{2} - 1 : - (- u^{2} + 1)

= \int_{2}^{5} \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= - \int_{2}^{5} \frac{1}{- u ^{2} + 1} d u

Nutze das gemeinsame Integral :

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= - [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{2}^{5}

Vereinfache

- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{2}^{5} : - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{2}^{5}

= - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{2}^{5}

Berechne die Grenzen:

\frac{ln ( 5 + 1 ) - ln ( 5 - 1 ) - ln ( 2 + 1 ) + ln ( 2 - 1 )}{2}

= - \frac{ln ( 5 + 1 ) - ln ( 5 - 1 ) - ln ( 2 + 1 ) + ln ( 2 - 1 )}{2}

Vereinfache

= - \frac{ln ( 5 + 1 ) - ln ( ( 5 - 1 ) ( 2 + 1 ) ) + ln ( 2 - 1 )}{2}

\int_{1}^{3} 25 - (x - 5)^{2} d x

\int_{- 1}^{1} [(2 - x^{2}) - x^{2}] d x

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\int_{- 1}^{1} x^{3} (4 + x^{4})^{3} d x