integral of ln(x)(5x+3)

Lösung

\int ln (x) (5 x + 3) d x

\frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C

Schritte zur Lösung

\int ln (x) (5 x + 3) d x

Wende die partielle Integration an

= (5 x + 3) (x ln (x) - x) - \int 5 (x ln (x) - x) d x

\int 5 (x ln (x) - x) d x = 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2})

= (5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2})

Vereinfache

(5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2}) : \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C

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

\int (3 3 x^{4} - 3) d x

\int x^{2} (x^{3} + 5) d x

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

\int sin (22 x) cos (13 x) d x