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integral of sqrt(e^t-3)

Lösung

\int e^{t} - 3 d t

23 (- arctan (\frac{1}{3} (e^{t} - 3)) + \frac{1}{3} (e^{t} - 3)) + C

Schritte zur Lösung

\int e^{t} - 3 d t

Wende U-Substitution an

= \int \frac{u - 3}{u} d u

Wende U-Substitution an

= \int \frac{2 v ^{2}}{v ^{2} + 3} d v

Entferne die Konstante:

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

= 2 \cdot \int \frac{v ^{2}}{v ^{2} + 3} d v

\frac{v ^{2}}{v ^{2} + 3} = - \frac{3}{v ^{2} + 3} + 1

= 2 \cdot \int - \frac{3}{v ^{2} + 3} + 1 d v

Wende integrale Substitution an

= 2 \cdot \int 3 (- \frac{1}{w ^{2} + 1} + 1) d w

Entferne die Konstante:

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

= 23 \cdot \int - \frac{1}{w ^{2} + 1} + 1 d w

Wende die Summenregel an:

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

= 23 (- \int \frac{1}{w ^{2} + 1} d w + \int 1 d w)

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

\int 1 d w = w

= 23 (- arctan (w) + w)

Ersetze zurück

= 23 (- arctan (\frac{e ^{t} - 3}{3}) + \frac{e ^{t} - 3}{3})

Vereinfache

23 (- arctan (\frac{e ^{t} - 3}{3}) + \frac{e ^{t} - 3}{3}) : 23 (- arctan (\frac{1}{3} (e^{t} - 3)) + \frac{1}{3} (e^{t} - 3))

= 23 (- arctan (\frac{1}{3} (e^{t} - 3)) + \frac{1}{3} (e^{t} - 3))

Füge eine Konstante zur Lösung hinzu

= 23 (- arctan (\frac{1}{3} (e^{t} - 3)) + \frac{1}{3} (e^{t} - 3)) + C

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