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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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