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Beliebt Rechnen >

integral of (4x^2+x+2)/(x+1)

Lösung

\int \frac{4 x ^{2} + x + 2}{x + 1} d x

Lösung

2 x^{2} - 3 x + 5 ln ∣ x + 1 ∣ - 5 + C

Schritte zur Lösung

\int \frac{4 x ^{2} + x + 2}{x + 1} d x

Multipliziere aus

\frac{4 x ^{2} + x + 2}{x + 1} : \frac{4 x ^{2}}{x + 1} + \frac{x}{x + 1} + \frac{2}{x + 1}

Wende die Summenregel an:

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

= \int \frac{4 x ^{2}}{x + 1} d x + \int \frac{x}{x + 1} d x + \int \frac{2}{x + 1} d x

\int \frac{4 x ^{2}}{x + 1} d x = 4 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣)

\int \frac{x}{x + 1} d x = x + 1 - ln ∣ x + 1 ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

= 4 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣) + x + 1 - ln ∣ x + 1 ∣ + 2 ln ∣ x + 1 ∣

Vereinfache

4 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣) + x + 1 - ln ∣ x + 1 ∣ + 2 ln ∣ x + 1 ∣ : 2 x^{2} - 3 x + 5 ln ∣ x + 1 ∣ - 5

= 2 x^{2} - 3 x + 5 ln ∣ x + 1 ∣ - 5

Füge eine Konstante zur Lösung hinzu

= 2 x^{2} - 3 x + 5 ln ∣ x + 1 ∣ - 5 + C

Graph

Beliebte Beispiele

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