integral of xsqrt(x^2+2x)

Lösung

\int x x^{2} + 2 x d x

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

Schritte zur Lösung

\int x x^{2} + 2 x d x

Wende die partielle Integration an

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

\int \frac{x ^{2} ( x + 1 )}{2 x ^{2} + 2 x} d x = \frac{1}{6} (- 3 x x^{2} + 2 x + (x^{2} + 2 x)^{\frac{3}{2}} - 3 ln x + x^{2} + 2 x + 1 + 3 x^{2} + 2 x)

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

Füge eine Konstante zur Lösung hinzu

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

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