integral of (3x^2-x+2)/((x+3)^3)

Lösung

\int \frac{3 x ^{2} - x + 2}{( x + 3 ) ^{3}} d x

- \frac{3 x ^{2}}{2 ( x + 3 ) ^{2}} + 3 ln ∣ x + 3 ∣ + \frac{10}{x + 3} - \frac{3}{2 ( x + 3 ) ^{2}} - \frac{1}{( x + 3 ) ^{2}} + C

Schritte zur Lösung

\int \frac{3 x ^{2} - x + 2}{( x + 3 ) ^{3}} d x

Multipliziere aus

\frac{3 x ^{2} - x + 2}{( x + 3 ) ^{3}} : \frac{3 x ^{2}}{( x + 3 ) ^{3}} - \frac{x}{( x + 3 ) ^{3}} + \frac{2}{( x + 3 ) ^{3}}

= \int \frac{3 x ^{2}}{( x + 3 ) ^{3}} - \frac{x}{( x + 3 ) ^{3}} + \frac{2}{( x + 3 ) ^{3}} d x

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{3 x ^{2}}{( x + 3 ) ^{3}} d x - \int \frac{x}{( x + 3 ) ^{3}} d x + \int \frac{2}{( x + 3 ) ^{3}} d x

\int \frac{3 x ^{2}}{( x + 3 ) ^{3}} d x = 3 (- \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3})

\int \frac{x}{( x + 3 ) ^{3}} d x = - \frac{1}{x + 3} + \frac{3}{2 ( x + 3 ) ^{2}}

\int \frac{2}{( x + 3 ) ^{3}} d x = - \frac{1}{( x + 3 ) ^{2}}

= 3 (- \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3}) - (- \frac{1}{x + 3} + \frac{3}{2 ( x + 3 ) ^{2}}) - \frac{1}{( x + 3 ) ^{2}}

Vereinfache

3 (- \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3}) - (- \frac{1}{x + 3} + \frac{3}{2 ( x + 3 ) ^{2}}) - \frac{1}{( x + 3 ) ^{2}} : - \frac{3 x ^{2}}{2 ( x + 3 ) ^{2}} + 3 ln ∣ x + 3 ∣ + \frac{10}{x + 3} - \frac{3}{2 ( x + 3 ) ^{2}} - \frac{1}{( x + 3 ) ^{2}}

= - \frac{3 x ^{2}}{2 ( x + 3 ) ^{2}} + 3 ln ∣ x + 3 ∣ + \frac{10}{x + 3} - \frac{3}{2 ( x + 3 ) ^{2}} - \frac{1}{( x + 3 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{3 x ^{2}}{2 ( x + 3 ) ^{2}} + 3 ln ∣ x + 3 ∣ + \frac{10}{x + 3} - \frac{3}{2 ( x + 3 ) ^{2}} - \frac{1}{( x + 3 ) ^{2}} + C

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