integral of 2xln(5x)

Lösung

\int 2 x ln (5 x) d x

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

Schritte zur Lösung

\int 2 x ln (5 x) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x ln (5 x) d x

Wende die partielle Integration an

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d x} = x^{4} y

\int \frac{1}{( y + 4 ) ^{2}} d y

x \to \infty lim (4 x + 1)

d er i v a t i v e 3 - x^{2}

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