integral of x(x-8)^{3/4}

Lösung

\int x (x - 8)^{\frac{3}{4}} d x

\frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}} + C

Schritte zur Lösung

\int x (x - 8)^{\frac{3}{4}} d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \int \frac{3 x ^{2}}{8 ( x - 8 ) ^{\frac{1}{4}}} d x

\int \frac{3 x ^{2}}{8 ( x - 8 ) ^{\frac{1}{4}}} d x = \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}}

= \frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}}

Vereinfache

\frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}} : \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}}

= \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}} + C

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