integral of (xln(x^2+1)+(x^2+1)y)

Lösung

\int (x ln (x^{2} + 1) + (x^{2} + 1) y) d x

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

Schritte zur Lösung

\int x ln (x^{2} + 1) + (x^{2} + 1) y d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d X} (800 X - 200)

\int \frac{1}{2} (3 x - 3)^{8} d x

\int_{0}^{π} x^{3} sin (x) d x

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

x \to - 3 - lim (\frac{∣ x - 3 ∣}{x - 3})