integral of x^25^x

Lösung

\int x^{2} 5^{x} d x

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

Schritte zur Lösung

\int x^{2} \cdot 5^{x} d x

Wende die partielle Integration an

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{10}{x} d x

x \to 0 lim (\frac{3}{x ^{2} + x} - \frac{3}{x})

\int_{- 1}^{1} \frac{2}{1 + x ^{2}} d x

\frac{d x}{d t} = 12 - \frac{27 x}{900 - 15 t}

\int \frac{sin ( x )}{1 + sin ^{2} ( x )} d x