integral of (e^x)/(sqrt(1-25e^{2x))}

Lösung

\int \frac{e ^{x}}{1 - 25 e ^{2 x}} d x

i \frac{1}{5} ln tan (arcsec (1 - 25 e^{2 x})) + sec (arcsec (1 - 25 e^{2 x})) + C

Schritte zur Lösung

\int \frac{e ^{x}}{1 - 25 e ^{2 x}} d x

Wende U-Substitution an

= \int i \frac{- 1 + u ^{2}}{- 5 + 5 u ^{2}} d u

Entferne die Konstante:

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

= i \cdot \int \frac{- 1 + u ^{2}}{- 5 + 5 u ^{2}} d u

Vereinfache

\frac{- 1 + u ^{2}}{- 5 + 5 u ^{2}} : \frac{1}{5 - 1 + u ^{2}}

= i \cdot \int \frac{1}{5 - 1 + u ^{2}} d u

Entferne die Konstante:

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

= i \frac{1}{5} \cdot \int \frac{1}{- 1 + u ^{2}} d u

Trigonometrische Substitution anwenden

= i \frac{1}{5} \cdot \int sec (v) d v

Nutze das gemeinsame Integral :

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= i \frac{1}{5} ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= i \frac{1}{5} ln tan (arcsec (1 - 25 e^{2 x})) + sec (arcsec (1 - 25 e^{2 x}))

Füge eine Konstante zur Lösung hinzu

= i \frac{1}{5} ln tan (arcsec (1 - 25 e^{2 x})) + sec (arcsec (1 - 25 e^{2 x})) + C

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