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

Lösung

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

n \to \infty lim (∣ n ∣)

\int e^{\frac{4}{x}} d x

a re a y = x, y = \frac{1}{x ^{2}}, x = 2

y^{^{'}} + \frac{2 y}{x} = - \frac{4}{x}, y (1) = 3

(\frac{1}{e ^{x}})^{^{'}}