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Beliebt Rechnen >

integral of x/(x+sqrt(x^2+1))

Lösung

\int \frac{x}{x + x ^{2} + 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x}{x + x ^{2} + 1} d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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