integral of x^22^x

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

= \frac{2 ^{x} x ^{2}}{ln ( 2 )} - \frac{1}{ln ( 2 )} (\frac{2 ^{x + 1} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{1 + x})

Füge eine Konstante zur Lösung hinzu

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

n = 0 \sum \infty \frac{x ^{n}}{3 ^{n}}

\int \frac{x ^{5} + x}{x + 1} d x

d er i v a t i v e x + 8 x

x \to \infty lim (\frac{e ^{4 x}}{5 x})

d er i v a t i v e (sin (x))^{(87)}