integral of (xln(x))^2

Lösung

\int (x ln (x))^{2} d x

\frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

Schritte zur Lösung

\int (x ln (x))^{2} d x

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

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

Wende die partielle Integration an

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

\int \frac{2}{3} x^{2} ln (x) d x = \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

= \frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

x \to 0 lim (x sin (\frac{7 π}{x}))

n or ma l 2 x^{2} - 3 x + 6, at x = 2

s l o p e x^{2}

y^{^{'}} y - x^{2} y = - x y

\int tan^{5} (x) sec^{3} (x) d x