integral from 3 to 5 of x/(x^2+4x+13)

Lösung

\int_{3}^{5} \frac{x}{x ^{2} + 4 x + 13} d x

\frac{1}{2} ln (\frac{29}{17}) - \frac{2}{3} arctan (\frac{7}{3}) + \frac{2}{3} arctan (\frac{5}{3})

Dezimale

0.17668 \dots

Schritte zur Lösung

\int_{3}^{5} \frac{x}{x ^{2} + 4 x + 13} d x

Vervollständige das Quadrat

x^{2} + 4 x + 13 : (x + 2)^{2} + 9

= \int_{3}^{5} \frac{x}{( x + 2 ) ^{2} + 9} d x

Wende U-Substitution an

= \int_{5}^{7} \frac{u - 2}{u ^{2} + 9} d u

Wende integrale Substitution an

= \int_{\frac{5}{3}}^{\frac{7}{3}} \frac{3 v - 2}{3 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int_{\frac{5}{3}}^{\frac{7}{3}} \frac{3 v - 2}{v ^{2} + 1} d v

Multipliziere aus

\frac{3 v - 2}{v ^{2} + 1} : \frac{3 v}{v ^{2} + 1} - \frac{2}{v ^{2} + 1}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int_{\frac{5}{3}}^{\frac{7}{3}} \frac{3 v}{v ^{2} + 1} d v - \int_{\frac{5}{3}}^{\frac{7}{3}} \frac{2}{v ^{2} + 1} d v)

\int_{\frac{5}{3}}^{\frac{7}{3}} \frac{3 v}{v ^{2} + 1} d v = \frac{3}{2} ln (\frac{29}{17})

\int_{\frac{5}{3}}^{\frac{7}{3}} \frac{2}{v ^{2} + 1} d v = 2 (arctan (\frac{7}{3}) - arctan (\frac{5}{3}))

= \frac{1}{3} (\frac{3}{2} ln (\frac{29}{17}) - 2 (arctan (\frac{7}{3}) - arctan (\frac{5}{3})))

Vereinfache

\frac{1}{3} (\frac{3}{2} ln (\frac{29}{17}) - 2 (arctan (\frac{7}{3}) - arctan (\frac{5}{3}))) : \frac{1}{2} ln (\frac{29}{17}) - \frac{2}{3} arctan (\frac{7}{3}) + \frac{2}{3} arctan (\frac{5}{3})

= \frac{1}{2} ln (\frac{29}{17}) - \frac{2}{3} arctan (\frac{7}{3}) + \frac{2}{3} arctan (\frac{5}{3})

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