integral of (e^x-5)^2

Lösung

\int (e^{x} - 5)^{2} d x

\frac{1}{2} e^{2 x} - 10 e^{x} + 25 x + C

Schritte zur Lösung

\int (e^{x} - 5)^{2} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 5 ) ^{2}}{u} : u - 10 + \frac{25}{u}

= \int u - 10 + \frac{25}{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 u d u - \int 10 d u + \int \frac{25}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= \frac{u ^{2}}{2} - 10 u + 25 ln ∣ u ∣

Setze in

u = e^{x}

ein

= \frac{( e ^{x} ) ^{2}}{2} - 10 e^{x} + 25 ln ∣ e^{x} ∣

Vereinfache

\frac{( e ^{x} ) ^{2}}{2} - 10 e^{x} + 25 ln ∣ e^{x} ∣ : \frac{1}{2} e^{2 x} - 10 e^{x} + 25 x

= \frac{1}{2} e^{2 x} - 10 e^{x} + 25 x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} e^{2 x} - 10 e^{x} + 25 x + C

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