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

Lösung

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

- \frac{ln ( 2 ) - ln ( 3 )}{2}

Dezimale Notation

0.20273 \dots

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

: \int_{2}^{3} \frac{1}{u ^{2} - 1} d u

= \int_{2}^{3} \frac{1}{u ^{2} - 1} d u

Faktorisiere

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

= \int_{2}^{3} \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}^{3} \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}^{3}

Vereinfache

- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{2}^{3} : - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{2}^{3}

= - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{2}^{3}

Berechne die Grenzen:

\frac{ln ( 2 ) - ln ( 3 )}{2}

= - \frac{ln ( 2 ) - ln ( 3 )}{2}

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

\int_{0}^{1} 6 π x d x

\int_{0}^{\infty} e^{- 3 t} d t

\int_{0}^{2} \frac{1}{ln ( x )} d x

\int_{0}^{l n (3)} 6 e^{3 x} d x