integral from 0 to 5 of 1/(x^2+6x+11)

Lösung

\int_{0}^{5} \frac{1}{x ^{2} + 6 x + 11} d x

\frac{arctan ( 4 2 ) - arctan ( \frac{3}{2} )}{2}

Dezimale

0.18776 \dots

Schritte zur Lösung

\int_{0}^{5} \frac{1}{x ^{2} + 6 x + 11} d x

Vervollständige das Quadrat

x^{2} + 6 x + 11 : (x + 3)^{2} + 2

= \int_{0}^{5} \frac{1}{( x + 3 ) ^{2} + 2} d x

Wende U-Substitution an

= \int_{3}^{8} \frac{1}{u ^{2} + 2} d u

Wende integrale Substitution an

= \int_{\frac{3}{2}}^{42} \frac{1}{2 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{\frac{3}{2}}^{42} \frac{1}{v ^{2} + 1} d v

Nutze das gemeinsame Integral :

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} [arctan (v)]_{\frac{3}{2}}^{42}

Berechne die Grenzen:

arctan (42) - arctan (\frac{3}{2})

= \frac{1}{2} (arctan (42) - arctan (\frac{3}{2}))

Vereinfache

= \frac{arctan ( 4 2 ) - arctan ( \frac{3}{2} )}{2}