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Beliebt Rechnen >

integral of x^32^x

Lösung

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

Lösung

\frac{1}{ln ^{4} ( 2 )} (ln^{3} (2) \cdot 2^{x} x^{3} - 3 (ln^{2} (2) \cdot 2^{x} x^{2} - 2 (ln (2) \cdot 2^{x} x - 2^{x}))) + C

Schritte zur Lösung

\int x^{3} \cdot 2^{x} d x

Wende U-Substitution an

= \int \frac{ln ^{3} ( u )}{ln ^{4} ( 2 )} d u

Entferne die Konstante:

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

= \frac{1}{ln ^{4} ( 2 )} \cdot \int ln^{3} (u) d u

Wende die partielle Integration an

= \frac{1}{ln ^{4} ( 2 )} (u ln^{3} (u) - \int 3 ln^{2} (u) d u)

\int 3 ln^{2} (u) d u = 3 (u ln^{2} (u) - 2 (u ln (u) - u))

= \frac{1}{ln ^{4} ( 2 )} (u ln^{3} (u) - 3 (u ln^{2} (u) - 2 (u ln (u) - u)))

Setze in

u = 2^{x}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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