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Beliebt Rechnen >

integral of (e^x2^{e^x}+2xe^{x^2})

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2 ^{u} u + 2 e ^{l n^{2} (u)} ln ( u )}{u} d u

Multipliziere aus

\frac{2 ^{u} u + 2 e ^{l n^{2} (u)} ln ( u )}{u} : 2^{u} + \frac{2 e ^{l n^{2} (u)} ln ( u )}{u}

= \int 2^{u} + \frac{2 e ^{l n^{2} (u)} ln ( 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

= \int 2^{u} d u + \int \frac{2 e ^{l n^{2} (u)} ln ( u )}{u} d u

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

\int \frac{2 e ^{l n^{2} (u)} ln ( u )}{u} d u = e^{l n^{2} (u)}

= \frac{2 ^{u}}{ln ( 2 )} + e^{l n^{2} (u)}

Setze in

u = e^{x}

ein

= \frac{2 ^{e^{x}}}{ln ( 2 )} + e^{l n^{2} (e^{x})}

Vereinfache

ln (e^{x}) : x

= \frac{2 ^{e^{x}}}{ln ( 2 )} + e^{x^{2}}

Füge eine Konstante zur Lösung hinzu

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

Graph

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