integral of x^24^x

Lösung

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

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

Schritte zur Lösung

\int x^{2} \cdot 4^{x} d x

Wende die partielle Integration an

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{4 x ^{2}}{9 - 4 x ^{2}} d x

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

\int x^{5} 9 + x^{6} d x

\int \frac{t ^{2} + 2}{t ^{2}} d t