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Beliebt Rechnen >

integral of (2^x+x^2)^2

Lösung

\int (2^{x} + x^{2})^{2} d x

Lösung

\frac{1}{5 ln ^{3} ( 2 )} (5 ln^{2} (2) \cdot 2^{2 x - 1} + 5 \cdot 2^{x + 1} (ln^{2} (2) x^{2} - 2 (ln (2) x - 1)) + ln^{3} (2) x^{5}) + C

Schritte zur Lösung

\int (2^{x} + x^{2})^{2} d x

Wende U-Substitution an

= \int \frac{( ln ^{2} ( 2 ) u + ln ^{2} ( u ) ) ^{2}}{ln ^{5} ( 2 ) u} d u

Entferne die Konstante:

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

= \frac{1}{ln ^{5} ( 2 )} \cdot \int \frac{( ln ^{2} ( 2 ) u + ln ^{2} ( u ) ) ^{2}}{u} d u

Multipliziere aus

\frac{( ln ^{2} ( 2 ) u + ln ^{2} ( u ) ) ^{2}}{u} : ln^{4} (2) u + 2 ln^{2} (2) ln^{2} (u) + \frac{ln ^{4} ( u )}{u}

= \frac{1}{ln ^{5} ( 2 )} \cdot \int ln^{4} (2) u + 2 ln^{2} (2) ln^{2} (u) + \frac{ln ^{4} ( u )}{u} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{ln ^{5} ( 2 )} (\int ln^{4} (2) u d u + \int 2 ln^{2} (2) ln^{2} (u) d u + \int \frac{ln ^{4} ( u )}{u} d u)

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

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

\int \frac{ln ^{4} ( u )}{u} d u = \frac{ln ^{5} ( u )}{5}

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

Setze in

u = 2^{x}

ein

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

Vereinfache

\frac{1}{ln ^{5} ( 2 )} (\frac{ln ^{4} ( 2 ) ( 2 ^{x} ) ^{2}}{2} + 2 ln^{2} (2) (2^{x} ln^{2} (2^{x}) - 2 (2^{x} ln (2^{x}) - 2^{x})) + \frac{ln ^{5} ( 2 ^{x} )}{5}) : \frac{1}{5 ln ^{3} ( 2 )} (5 ln^{2} (2) \cdot 2^{2 x - 1} + 5 \cdot 2^{x + 1} (ln^{2} (2) x^{2} - 2 (ln (2) x - 1)) + ln^{3} (2) x^{5})

= \frac{1}{5 ln ^{3} ( 2 )} (5 ln^{2} (2) \cdot 2^{2 x - 1} + 5 \cdot 2^{x + 1} (ln^{2} (2) x^{2} - 2 (ln (2) x - 1)) + ln^{3} (2) x^{5})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5 ln ^{3} ( 2 )} (5 ln^{2} (2) \cdot 2^{2 x - 1} + 5 \cdot 2^{x + 1} (ln^{2} (2) x^{2} - 2 (ln (2) x - 1)) + ln^{3} (2) x^{5}) + C

Graph

Beliebte Beispiele

integral of (2x^2)/(x-1)

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

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\int \frac{x ^{2}}{x ^{2} - 81} d x

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\int 5 sin (6 x) d x