integral of ln(xe^x)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

= x (ln (x) + x) - \int 1 + x d x

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x ^{2}}{1 - 4 x ^{2}} d x

\int (12 x^{\frac{3}{4}} - 9 x^{\frac{5}{3}}) d x

\int \frac{5 x + 1}{x ^{2} - 6 x + 5} d x

\int \frac{3 x ^{2} + 1}{x} d x

\int ∣ x ∣ - 1 d x