integral from 3 to 6 of (12x)/(x^2+2x-8)

Lösung

\int_{3}^{6} \frac{12 x}{x ^{2} + 2 x - 8} d x

2 (3 ln (40) + ln (\frac{5}{2}) - 4 ln (7))

Dezimale

8.39857 \dots

Schritte zur Lösung

\int_{3}^{6} \frac{12 x}{x ^{2} + 2 x - 8} d x

Entferne die Konstante:

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

= 12 \cdot \int_{3}^{6} \frac{x}{x ^{2} + 2 x - 8} d x

Vervollständige das Quadrat

x^{2} + 2 x - 8 : (x + 1)^{2} - 9

= 12 \cdot \int_{3}^{6} \frac{x}{( x + 1 ) ^{2} - 9} d x

Wende U-Substitution an

= 12 \cdot \int_{4}^{7} \frac{u - 1}{u ^{2} - 9} d u

Multipliziere aus

\frac{u - 1}{u ^{2} - 9} : \frac{u}{u ^{2} - 9} - \frac{1}{u ^{2} - 9}

Wende die Summenregel an:

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

= 12 (\int_{4}^{7} \frac{u}{u ^{2} - 9} d u - \int_{4}^{7} \frac{1}{u ^{2} - 9} d u)

\int_{4}^{7} \frac{u}{u ^{2} - 9} d u = \frac{1}{2} (ln (40) - ln (7))

\int_{4}^{7} \frac{1}{u ^{2} - 9} d u = - \frac{ln ( \frac{5}{2} ) - ln ( 7 )}{6}

= 12 (\frac{1}{2} (ln (40) - ln (7)) - (- \frac{ln ( \frac{5}{2} ) - ln ( 7 )}{6}))

Vereinfache

12 (\frac{1}{2} (ln (40) - ln (7)) - (- \frac{ln ( \frac{5}{2} ) - ln ( 7 )}{6})) : 2 (3 ln (40) + ln (\frac{5}{2}) - 4 ln (7))

= 2 (3 ln (40) + ln (\frac{5}{2}) - 4 ln (7))

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