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Beliebt Rechnen >

integral of (e^{4x})/(sqrt(25-e^{2x))}

Lösung

\int \frac{e ^{4 x}}{25 - e ^{2 x}} d x

Lösung

- \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25 + C

Schritte zur Lösung

\int \frac{e ^{4 x}}{25 - e ^{2 x}} d x

Wende U-Substitution an

= \int - \frac{- u + 25}{2 u} d u

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int \frac{- u + 25}{u} d u

Multipliziere aus

\frac{- u + 25}{u} : - u + \frac{25}{u}

= - \frac{1}{2} \cdot \int - u + \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

= - \frac{1}{2} (- \int u d u + \int \frac{25}{u} d u)

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{25}{u} d u = 50 u

= - \frac{1}{2} (- \frac{2}{3} u^{\frac{3}{2}} + 50 u)

Setze in

u = 25 - e^{2 x}

ein

= - \frac{1}{2} (- \frac{2}{3} (25 - e^{2 x})^{\frac{3}{2}} + 50 25 - e^{2 x})

Vereinfache

- \frac{1}{2} (- \frac{2}{3} (25 - e^{2 x})^{\frac{3}{2}} + 50 25 - e^{2 x}) : - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25

= - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25 + C

Graph

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