integral of (x^3+2x-1)/((x-3)^3)

Lösung

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

- \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x ^{3}}{( x - 3 ) ^{3}} d x = - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )}

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

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

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} - (- \frac{1}{2 ( x - 3 ) ^{2}})

Vereinfache

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}} + C

\int \frac{4 x ^{3}}{x ^{2} - 9} d x

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\int r^{2} r d r