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Beliebt Rechnen >

integral of (x^3-4x+6)/(x+7)

Lösung

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

Lösung

x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6 + C

Schritte zur Lösung

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

Wende U-Substitution an

= \int \frac{- 4 u + ( u - 7 ) ^{3} + 34}{u} d u

Wende die partielle Integration an

= - 4 u + (u - 7)^{3} + 34 - \int \frac{3 u ^{3} - 42 u ^{2} + 147 u - ( u - 7 ) ^{3} - 34}{u} d u

\int \frac{3 u ^{3} - 42 u ^{2} + 147 u - ( u - 7 ) ^{3} - 34}{u} d u = \frac{2 u ^{3}}{3} - \frac{21 u ^{2}}{2} + 309 ln ∣ u ∣

= - 4 u + (u - 7)^{3} + 34 - (\frac{2 u ^{3}}{3} - \frac{21 u ^{2}}{2} + 309 ln ∣ u ∣)

Setze in

u = x + 7

ein

= - 4 (x + 7) + (x + 7 - 7)^{3} + 34 - (\frac{2 ( x + 7 ) ^{3}}{3} - \frac{21 ( x + 7 ) ^{2}}{2} + 309 ln ∣ x + 7 ∣)

Vereinfache

- 4 (x + 7) + (x + 7 - 7)^{3} + 34 - (\frac{2 ( x + 7 ) ^{3}}{3} - \frac{21 ( x + 7 ) ^{2}}{2} + 309 ln ∣ x + 7 ∣) : x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6

= x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6

Füge eine Konstante zur Lösung hinzu

= x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6 + C

Graph

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