integral of x^3(x^2-1)^{3/2}

Lösung

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

\frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}} + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \int \frac{3}{4} x^{5} (x^{2} - 1)^{\frac{1}{2}} d x

\int \frac{3}{4} x^{5} (x^{2} - 1)^{\frac{1}{2}} d x = \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} + \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} + \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}}

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - (\frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} + \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} + \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}})

Vereinfache

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{4} (x^{2} - 1)^{\frac{3}{2}} - \frac{3}{10} (x^{2} - 1)^{\frac{5}{2}} - \frac{3}{28} (x^{2} - 1)^{\frac{7}{2}} + C

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