integral of x(1+x^2)^{5/2}

Lösung

\int x (1 + x^{2})^{\frac{5}{2}} d x

\frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

Schritte zur Lösung

\int x (1 + x^{2})^{\frac{5}{2}} d x

Wende die partielle Integration an

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

\int \frac{5}{2} x^{3} (1 + x^{2})^{\frac{3}{2}} d x = \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

Vereinfache

\frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}} : \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

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